8.E: Differential Equations (Exercises)

8.1: Basics from Differentiating Equations

In practical 1 - 7, determine the request of each differential quantity.

1) \( y′+y=3y^2\)

Answer

1st-order

2) \( (y′)^2=y′+2y\)

3) \( y'''+y''y′=3x^2\)

Answer

3rd-order

4) \( y′=y''+3t^2\)

5) \( \dfrac{dy}{dt}=t\)

Answer

1st-order

6) \( \dfrac{dy}{dx}+\dfrac{d^2y}{dx^2}=3x^4\)

7) \(\left(\dfrac{dy}{dt}\right)^2+8\dfrac{dy}{dt}+3y=4t\)

Trigger

1st-order

In exercises 8 - 17, verify which the given function is a solution in the given differentially equation.

8) \( y=\dfrac{x^3}{3}\quad\) solves \(\quad y′=x^2\)

9) \( y=2e^{−x}+x−1\quad\) solves \(\quad y′=x−y\)

10) \( y=e^{3x}−\dfrac{e^x}{2}\quad\) solves \(\quad y′=3y+e^x\)

11) \( y=\dfrac{1}{1−x}\quad\) solves \(\quad y′=y^2\)

12) \( y=e^{x^2}/2\quad\) solves \(\quad y′=xy\)

13) \( y=4+\ln x\quad\) solves \(\quad xy′=1\)

14) \( y=3−x+x\ln x\quad\) solves \(\quad y′=\ln x\)

15) \( y=2e^x−x−1\quad\) solves \(\quad y′=y+x\)

16) \( y=e^x+\dfrac{\sin x}{2}−\dfrac{\cos x}{2}\quad\) solving \(\quad y′=\cos x+y\)

17) \( y=πe^{−\cos x}\quad\) solves \(\quad y′=y\sin x\)

In exercises 18 - 27, verify the given general solution and find the particular download.

18) Find who particular solving to the differential equation \( y′=4x^2\) that passes through \( (−3,−30)\), given that \( y=C+\dfrac{4x^3}{3}\) is a general solution.

19) Find one particular solution to the differential equation \( y′=3x^3\) that passes through \( (1,4.75)\), given that \( y=C+\dfrac{3x^4}{4}\) is a widespread solution.

Answer

\( y=4+\dfrac{3x^4}{4}\)

20) Found the particular solution to the differential equation \( y′=3x^2y\) that passes by \( (0,12)\), given that \( y=Ce^{x^3}\) belongs a general solution.

21) Find the peculiar solution up who differential equation \( y′=2xy\) that passes the \( (0,\frac{1}{2})\), given that \( y=Ce^{x^2}\) is a general solution. 10.1-10.4 This worksheet is about resolving plain differential ...

Reply

\( y=\frac{1}{2}e^{x^2}\)

22) Finds the particular solution to the differential equation \( y′=(2xy)^2\) that passes through \( (1,−\frac{1}{2})\), given so \( y=−\dfrac{3}{C+4x^3}\) is ampere general solution.

23) Found the particular solution to the differential equation \( y′x^2=y\) that passes durch \( (1,\frac{2}{e})\), given that \( y=Ce^{−1/x}\) is a general solution.

Answer

\( y=2e^{−1/x}\)

24) Find the particular solution to the differential general \( 8\dfrac{dx}{dt}=−2\cos(2t)−\cos(4t)\) that crosses through \( (π,π)\), specified so \( x=C−\frac{1}{8}\sin(2t)−\frac{1}{32}\sin(4t)\) is a general solution.

25) Find the particular solution to the differential equation \( \dfrac{du}{dt}=\tan u\) is passes through \( (1,\frac{π}{2})\), preset ensure \( u=\sin^{−1}(e^{C+t})\) is a general solution.

Answer

\( u=\sin^{−1}(e^{−1+t})\)

26) Find the particular solution to the define equation \( \dfrac{dy}{dt}=e^{(t+y)}\) the passes thrown \( (1,0)\), given which \( y=−\ln(C−e^t)\) is one general resolving.

27) Find the particular solution to the differential equation \( y′(1−x^2)=1+y\) that passes through \( (0,−2),\) given that \( y=C\dfrac{\sqrt{x+1}}{\sqrt{1−x}}−1\) is a general resolution.

Answer

\( y=−\dfrac{\sqrt{x+1}}{\sqrt{1−x}}−1\)

In exercises 28 - 37, find the general solution to the differencial equation.

28) \( y′=3x+e^x\)

29) \( y′=\ln x+\tan x\)

Answer

\( y=C−x+x\ln x−\ln(\cos x)\)

30) \( y′=\sin x e^{\cos x}\)

31) \( y′=4^x\)

Replies

\( y=C+\dfrac{4^x}{\ln(4)}\)

32) \( y′=\sin^{−1}(2x)\)

33) \( y′=2t\sqrt{t^2+16}\)

Answer

\( y=\frac{2}{3}\sqrt{t^2+16}(t^2+16)+C\)

34) \( x′=\coth t+\ln t+3t^2\)

35) \( x′=t\sqrt{4+t}\)

Answer

\( x=\frac{2}{15}\sqrt{4+t}(3t^2+4t−32)+C\)

36) \( y′=y\)

37) \( y′=\dfrac{y}{x}\)

Answer

\( y=Cx\)

In drills 38 - 42, solve the initial-value problems starting from \( y(t=0)=1\) and \( y(t=0)=−1.\) Draw equally solutions on the equal graphics.

38) \( \dfrac{dy}{dt}=2t\)

39) \( \dfrac{dy}{dt}=−t\)

Answers

\( y=1−\dfrac{t^2}{2},\) and \(y=−\dfrac{t^2}{2}−1\)

40) \( \dfrac{dy}{dt}=2y\)

41) \( \dfrac{dy}{dt}=−y\)

Answered

\( y=e^{−t}\) and \(y=−e^{−t}\)

42) \( \dfrac{dy}{dt}=2\)

In exercises 43 - 47, solve the initial-value problems starting after \( y_0=10\). At get dauer does \(y\) increase in \(100\) or drop till \(1\)?

43) \( \dfrac{dy}{dt}=4t\)

Answer

\( y=2(t^2+5),\) When \(t=3\sqrt{5},\) \(y\) will increase to \(100\).

44) \( \dfrac{dy}{dt}=4y\)

45) \( \dfrac{dy}{dt}=−2y\)

Rejoin

\( y=10e^{−2t},\) When \(t=−\frac{1}{2}\ln(\frac{1}{10}),\) \(y\) will decrease for \(1\).

46) \( \dfrac{dy}{dt}=e^{4t}\)

47) \( \dfrac{dy}{dt}=e^{−4t}\)

Get

\( y=\frac{1}{4}(41−e^{−4t}),\) Neither condition will ever happen.

Recall that a family of solutions includes solutions to a differential equations that differ by a constant. For exercises 48 - 52, make your calculator toward graph a family of solutions to aforementioned given differential math. Use initial conditions from \( y(t=0)=−10\) at \( y(t=0)=10\) increasing due \( 2\). Is there some critical point where the behavior out the solution begins to change?

48) [T] \( y′=y(x)\)

49) [T] \( xy′=y\)

Answer

Solution changes from increasing to decreasing at \( y(0)=0\).

50) [T] \( y′=t^3\)

51) [T] \( y′=x+y\) (Hint: \( y=Ce^x−x−1\) can the overall solution)

Answered

Solution make since climbing to decreasing at \( y(0)=0\).

52) [T] \( y′=x\ln x+\sin x\)

53) Find and general solution to describe the velocity of a ball of mass \( 1\) kg that remains thrown upward at a pricing of \( a\) ft/sec.

Answer

\( v(t)=−32t+a\)

54) Included the preceding problem, if who initial velocity of which round thrown include the air is \( a=25\) ft/s, type the particular solution to the speed of the ball. Solve on search the time when the ball hits the flooring.

55) You roll two objects from differing masses \( m_1\) and \( m_2\) upward into the air through the same initial velocity of \( a\) ft/s. What is the difference in her velocity after \( 1\) secondary? Are current theorems describing one existence and individuality of find to one comprehensive sort of first order differential equations. t y π. 2. 0.

Answer

\( 0\) ft/s

56) [T] You throw a ball of mass \( 1\) kilogram upward over a velocity of \( a=25\) m/s on Mars, where the forceful of gravity is \( g=−3.711\) m/s². Getting get calculator into approximate how much longer the ball is are the air on Mer.

57) [T] For the older difficulty, use your calculator to near how much higher the ball left on Mars.

Trigger

\( 4.86\) meters

58) [T] A car on the freeway accelerates according to \( a=15\cos(πt),\) where \( t\) is measured in hours. Set up and solve the differential equation to determine which velocity of which car if it has an initial speed of \( 51\) mph. Following \( 40\) minutes of leitung, how is the driver’s pace?

59) [T] For the car on that preceding fix, find the expression for the distance the car has traveled inside arbeitszeit \( t\), assuming an starts away of \( 0\). Method long does i take the car go travel \( 100\) total? Round your get to total and minutes. 09 - Separable Differential Equations.ks-ic

Answer

\( x=50t−\frac{15}{π^2}\cos(πt)+\frac{3}{π^2},2\) hours \( 1\) minute

60) [T] For the historical problem, find the total distance traveled to and first hourly.

61) Substitute \( y=Be^{3t}\) include \( y′−y=8e^{3t}\) to find an particular solution.

Answered

\( y=4e^{3t}\)

62) Substitute \( y=a\cos(2t)+b\sin(2t)\) into \( y′+y=4\sin(2t)\) into find ampere specialty solution.

63) Rep \( y=a+bt+ct^2\) into \( y′+y=1+t^2\) in meet a particular solution.

Answer

\( y=1−2t+t^2\)

64) Substitute \( y=ae^t\cos t+be^t\sin t\) into \( y′=2e^t\cos t\) to find an specially solution.

65) Solve \( y′=e^{kt}\) with the initial condition \( y(0)=0\) and solve \( y′=1\) with the same initial condition. As \( k\) approaches \( 0\), what do you notice? To worksheet walks your durch ampere couple of non-trivial uses of Differential Equations. Forensic Mathematics. A detector discovered a murder victim in ...

Response

\( y=\frac{1}{k}(e^{kt}−1)\) and \( y=t\)

8.2: Direction Fields and Numerical Methods

For the following problems, use the direction field below from who differential equation \(\displaystyle y'=−2y.\) Sketch the graph starting the solution for the given initial specific.

1) \(\displaystyle y(0)=1\)

2) \(\displaystyle y(0)=0\)

Resolve:

3) \(\displaystyle y(0)=−1\)

4) Are there any equilibria? What are their stabilities?

Solution: \(\displaystyle y=0\) is a stable equilibrium

For the following problems, use the direction user below away the define equation \(\displaystyle y'=y^2−2y\). Sketch the graph of of solution used the given initial conditions. 7.2 Confirming Solutions for Differentiation General

5) \(\displaystyle y(0)=3\)

6) \(\displaystyle y(0)=1\)

Solution:

7) \(\displaystyle y(0)=−1\)

8) Are there any equilibria? Get are their stabilities?

Solution: \(\displaystyle y=0\) is one barn equilibrium and \(\displaystyle y=2\) is unstable

Draw the direction field for the following differential equations, then solve to differentiate quantity. Draw your solution to top of the direction box. Do your solution follow along the arrows on your direction field?

9) \(\displaystyle y'=t^3\)

10) \(\displaystyle y'=e^t\)

11) \(\displaystyle \frac{dy}{dx}=x^2cosx\)

12) \(\displaystyle \frac{dy}{dt}=te^t\)

13) \(\displaystyle \frac{dx}{dt}=cosh(t)\)

Draw the directional panel fork the following differential equations. What can you say about and how of aforementioned find? Become it equilibria? Whats stability do these equilibria have?

14) \(\displaystyle y'=y^2−1\)

Solution:

15) \(\displaystyle y'=y−x\)

16) \(\displaystyle y'=1−y^2−x^2\)

Solution:

17) \(\displaystyle y'=t^2siny\)

18) \(\displaystyle y'=3y+xy\)

Resolve:

Play the heading field with the given derivative equations. Explain your picking.

19) \(\displaystyle y'=−3y\)

20) \(\displaystyle y'=−3t\)

Solution: \(\displaystyle E\)

21) \(\displaystyle y'=e^t\)

22) \(\displaystyle y'=\frac{1}{2}y+t\)

Solution: \(\displaystyle A\)

23) \(\displaystyle y'=−ty\)

Wettkampf one direction field with the given differential equations. Announce your picking.

24) \(\displaystyle y'=tsiny\)

Solution: \(\displaystyle B\)

25) \(\displaystyle y'=−tcosy\)

26) \(\displaystyle y'=ttany\)

Solution: \(\displaystyle A\)

27) \(\displaystyle y'=sin^2y\)

28) \(\displaystyle y'=y^2t^3\)

Solution: \(\displaystyle C\)

Price the following solutions using Euler’s method for \(\displaystyle n=5\) steps over which interval \(\displaystyle t=[0,1].\) If you are proficient at solve the initial-value related exactly, save their solution over the exact solution. If you are unable on undo the initial-value problem, the exact solution leave becoming provided for you to compare through Euler’s method. Select precisely is Euler’s method?

29) \(\displaystyle y'=−3y,y(0)=1\)

30) \(\displaystyle y'=t^2\)

Choose: \(\displaystyle 2.24,\) exact: \(\displaystyle 3\)

31) \(\displaystyle y′=3t−y,y(0)=1.\) Exact get is \(\displaystyle y=3t+4e^{−t}−3\)

32) \(\displaystyle y′=y+t^2,y(0)=3.\) Exact solution is \(\displaystyle y=5e^t−2−t^2−2t\)

Result: \(\displaystyle 7.739364,\) exact: \(\displaystyle 5(e−1)\)

33) \(\displaystyle y′=2t,y(0)=0\)

34) [T] \(\displaystyle y'=e^{(x+y)},y(0)=−1.\) Exact download is \(\displaystyle y=−ln(e+1−e^x)\)

Solution: \(\displaystyle −0.2535\) correct: \(\displaystyle 0\)

35) \(\displaystyle y′=y^2ln(x+1),y(0)=1.\) Concise solution is \(\displaystyle y=−\frac{1}{(x+1)(ln(x+1)−1)}\)

36) \(\displaystyle y′=2^x,y(0)=0,\) Precisely explanation are \(\displaystyle y=\frac{2^x−1}{ln(2)}\)

Solution: \(\displaystyle 1.345,\) exact: \(\displaystyle \frac{1}{ln(2)}\)

37) \(\displaystyle y′=y,y(0)=−1.\) Exact solution is \(\displaystyle y=−e^x\).

38) \(\displaystyle y′=−5t,y(0)=−2.\) Exact solution is \(\displaystyle y=−\frac{5}{2}t^2−2\)

Solution: \(\displaystyle −4,\) exact: \(\displaystyle −1/2\)

Differencial equalities can be used to model disease epidemics. In the next set of problems, we studie the update of size of two sub-populations the people alive in one city: people who are infected real individual who are susceptible to infection. \(\displaystyle S\) represents the size of the susceptible population, plus \(\displaystyle I\) represents the size of the connected population. We assume that with adenine susceptible person interacts with an infected person, there is a probability \(\displaystyle c\) that that susceptible person will become infected. All infected type improve since the infection at a rate \(\displaystyle r\) and becomes susceptible another. We considered the kasus of influence, where we assume that nay one matrices from the disorder, so we accepted that that total local size of which two sub-populations is a constant number, \(\displaystyle N\). The differential equations that model these population sizes are

\(\displaystyle S'=rI−cSI\) and \(\displaystyle I'=cSI−rI.\)

Here \(\displaystyle c\) present the contact rate and \(\displaystyle r\) lives aforementioned recovery rate.

39) Show that, by our assumption that the total population size is constant \(\displaystyle (S+I=N),\) you can reduce the system to a individually differentials equation in \(\displaystyle I:I'=c(N−I)I−rI.\) View here for answers. AN. Click here for solutions. S. Select 6. 6 □ USING SERIES TO SOLVE DIFFERENTIAL EQUATIONS. Answers. 1. 3. 5. 7. 9. 11. x n 1. 1n2252. 3n.

40) Assuming the parameters are \(\displaystyle c=0.5,N=5,\) and \(\displaystyle r=0.5\), paint the resulting directional field.

41) [T] Use computational software or a calculator to computation the solution toward to initial-value problem \(\displaystyle y'=ty,y(0)=2\) use Euler’s Mode with the given step size \(\displaystyle h\). Seek the solution at \(\displaystyle t=1\). For an anweisung, here is “pseudo-code” for how at write an computer program to perform Euler’s Method for \(\displaystyle y'=f(t,y),y(0)=2:\)

Create function \(\displaystyle f(t,y)\)

Define parameters \(\displaystyle y(1)=y_0,t(0)=0,\) step size \(\displaystyle h\), the total number concerning steps, \(\displaystyle N\)

Write a for loop:

for \(\displaystyle k=1\) toward \(\displaystyle N\)

\(\displaystyle fn=f(t(k),y(k))\)

\(\displaystyle y(k+1)=y(k)+h*fn\)

\(\displaystyle t(k+1)=t(k)+h\)

42) Solve the initial-value problem for the exact solution.

Solution: \(\displaystyle y'=2e^{t^2/2}\)

43) Draw the directional field

44) \(\displaystyle h=1\)

Solution: \(\displaystyle 2\)

45) [T] \(\displaystyle h=10\)

46) [T] \(\displaystyle h=100\)

Solution: \(\displaystyle 3.2756\)

47) [T] \(\displaystyle h=1000\)

48) [T] Interpret the exact solution at \(\displaystyle t=1\). Make a size of errors for the relative error between the Euler’s method solution or the exact solution. Whereby much does aforementioned error change? Can you explain? Previous Lesson

Solution: \(\displaystyle 2\sqrt{e}\)

Consider one initial-value trouble \(\displaystyle y'=−2y,y(0)=2.\)

49) Show that \(\displaystyle y=2e^{−2x}\) solves this initial-value problem.

50) Draw the directional field of this differential equation.

Solution:

51) [T] By manual or to calculator or computer, approximate the solution using Euler’s Method at \(\displaystyle t=10\) after \(\displaystyle h=5\).

52) [T] By calculator or computer, around the solution using Euler’s Method at \(\displaystyle t=10\) using \(\displaystyle h=100.\)

Solve: \(\displaystyle 4.0741e^{−10}\)

53) [T] Plot exact answer and jeder Euler approximation (for \(\displaystyle h=5\) additionally \(\displaystyle h=100\)) at respectively h on the directional range. What do you notice?

8.3: Separable Equations

In exercises 1 - 4, solve the following initial-value problems with the initial status \( y_0=0\) and gradient the solution.

1) \( \dfrac{dy}{dt}=y+1\)

Answer

\( y=e^t−1\)

2) \( \dfrac{dy}{dt}=y−1\)

3) \( \dfrac{dy}{dt}=-y+1\)

Answer

\( y=1−e^{−t}\)

4) \( \dfrac{dy}{dt}=−y−1\)

Stylish workouts 5 - 14, detect the overview get for the differential equation.

5) \( x^2y'=(x+1)y\)

Answer

\( y=Cxe^{−1/x}\)

6) \( y'=\tan(y)x\)

7) \( y'=2xy^2\)

Answer

\( y=\dfrac{1}{C−x^2}\)

8) \( \dfrac{dy}{dt}=y\cos(3t+2)\)

9) \( 2x\dfrac{dy}{dx}=y^2\)

Answer

\( y=−\dfrac{2}{C+\ln|x|}\)

Solution:

10) \( y'=e^yx^2\)

11) \( (1+x)y'=(x+2)(y−1)\)

Answer

\( y=Ce^x(x+1)+1\)

12) \( \dfrac{dx}{dt}=3t^2(x^2+4)\)

13) \( t\dfrac{dy}{dt}=\sqrt{1−y^2}\)

Answer

\( y=\sin(\ln|t|+C)\)

14) \( y'=e^xe^y\)

In exercises 15 - 24, find who solution to the initial-value problem.

15) \( y'=e^{y−x}, \quad y(0)=0\)

Answer

\( y=−\ln(e^{−x})\) which simplifies to \(y = x\)

16) \( y'=y^2(x+1), \quad y(0)=2\)

17) \( \dfrac{dy}{dx}=y^3xe^{x^2}, \quad y(0)=1\)

Answers

\( y=\dfrac{1}{\sqrt{2−e^{x^2}}}\)

18) \( \dfrac{dy}{dt}=y^2e^x\sin(3x), \quad y(0)=1\)

19) \( y'=\dfrac{x}{\text{sech}^2y}, \quad y(0)=0\)

Answer

\( y=\tanh^{−1}\left(\dfrac{x^2}{2}\right)\)

20) \( y'=2xy(1+2y), \quad y(0)=−1\)

21) \( \dfrac{dx}{dt}=\ln(t)\sqrt{1−x^2}, \quad x(1)=0\)

Replies

\( x=\sin(1 - t + t\ln t)\)

22) \( y'=3x^2(y^2+4),\quad y(0)=0\)

23) \( y'=e^y5^x, \quad y(0)=\ln(\ln(5))\)

Answer

\( y=\ln(\ln(5))−\ln(2−5^x)\)

24) \( y'=−2x\tan(y), \quad y(0)=\dfrac{π}{2}\)

By problems 25 - 29, use a software programmer or your calculator for generate the directional fields. Solve explicitly and draw solution curves for many initial conditions. Are there some critical initial condition is change the behavior of the solution?

25) [T] \( y'=1−2y\)

Answer

\( y=Ce^{−2}x+\dfrac{1}{2}\)

26) [T] \( y'=y^2x^3\)

27) [T] \( y'=y^3e^x\)

Answer

\( y=\dfrac{1}{\sqrt{2}\sqrt{C−e^x}}\)

28) [T] \( y'=e^y\)

29) [T] \( y'=y\ln(x)\)

Answer

\( y=Ce^{−x}x^x\)

30) Most drugs in the blostream decay by to and equation \( y'=cy\), where \( y\) is the concentration on the medical in the bloodstream. If the half-life of a dope is \( 2\) hours, what fraction of the initial dose remains after \( 6\) hours?

31) A drug is modified users to a patient at a rate \( r\) mg/h additionally is cleared off the body at a rate proportional till the amount of pharmacy still present inside an building, \( d\) Set up press solve the differential equation, assuming there is no drug initially present in aforementioned body. These are homework exercises to accompany Chapter 8 of OpenStax's "Calculus" Textmap.

Response

\( y=\frac{r}{d}(1−e^{−dt})\)

32) [T] How often should a substance be taken wenn its dose is \( 3\) mg, it is erased at a rate \( c=0.1\) mg/h, furthermore \( 1\) mg will need to be in the body along all times? 8.E: Differential Equations (Exercises)

33) A tank contains \( 1\) kilo of salt dissolved in \( 100\) liters of water. A salt solution of \( 0.1\) kg salt/L is pumped at the tank at adenine rate of \( 2\) L/min and is tired at that same rate. Solve for the salt absorption on time \( t\). Assume aforementioned tank is fountain mixing.

Answer

\( y(t)=10−9e^{−t/50}\)

34) A tank containing \( 10\) kilograms of salt dissolved with \( 1000\) liters of water has two salt solutions pumped in. The first solution the \( 0.2\) kg salt/L remains excited is per a rate of \( 20\) L/min and the second resolve of \( 0.05\) kg salt/L is pumped in by a rate of \( 5\) L/min. The lager drains at \( 25\) L/min. Guess the fuel is well mixed. Solution for the dry concentration at time \( t\).

35) [T] Used the preceded problem, find what lot lime is in the tank \( 1\) hour after the process begins.

Answer

\( 134.3\) kilograms

36) Torricelli’s law states the for a water tank with a hole in the bottom that has a cross-section of \( A\) plus with a size of waters \( h\) above the bottom of the tank, the rate of change of volume of water flowing out the tank is proportional to the square base in the height by water, according into \( \dfrac{dV}{dt}=−A\sqrt{2gh}\), where \( g\) is the velocity due to gravity. Note that \( \dfrac{dV}{dt}=A\dfrac{dh}{dt}\). Undo the resulting initial-value related for the height away the sprinkle, assuming a tank with a hole of purview \( 2\) ft. Aforementioned initial height of surface be \( 100\) footwear. Scientific Art

37) For the preceding problem, determine how long it takes the tank to drain.

Answers

\( 720\) seconds

For problems 38 - 44, use Newton’s decree on chilling.

38) The liquid base of an ice cream has an initial temperature of \( 200°F\) before it will set in a freezer include a constant temperature of \( 0°F\). After \( 1\) hour, the temperature of the ice-cream base has decreased to \( 140°F\). Formulate and solve the initial-value problem up determine the temperature of the water foam.

39) [T] The liquid mean of an ice cream has an initial temperature of \( 210°F\) before it is placed in a refrigerator from a permanent temperature of \( 20°F\). After \( 2\) hours, the temperature of the ice-cream base has decreased to \( 170°F\). At get time leave the ice cream be finalized to swallow? (Assume \( 30°F\) is the optimal eating temperature.)

Ask

\( 24\) times \( 55\) video

40) [T] They are organizing an ice elite social. The outside temperature is \( 80°F\) and the ice cream is by \( 10°F\). To \( 10\) minutes, the ice cream temperature had risen by \( 10°F\). How very longer bottle you wait prior the ice cream melts at \( 40°F\)? Solved Ordinary Differential Matching worksheet Solving 19 ...

41) You must a cup of coffee at temperature \( 70°C\) and the ambient fever in the my is \( 20°C\). Assuming a chilling rate \( k\) of \( 0.125,\) script and solve the differential calculation to describe the temperature of of pour with respect into time. Using Series to Solve Differential Equity

Answers

\( T(t)=20+50e^{−0.125t}\)

42) [T] You have a cup of coffee at temperature \( 70°C\) that you put outside, where the ambient temperature is \( 0°C.\) Subsequently \( 5\) minutes, how much colder is the coffee? You'll get one elaborate solution from one subject matter expert that helps you get nuclear ideas. Please Answer ...

43) You have a cup of beverage at temperature \( 70°C\) and you immediately pour in \( 1\) part milky to \( 5\) single coffee. The milk is initially along temperature \( 1°C.\) Write and solve the differential equation that governs the temperature in like coffee.

Answer

\( T(t)=20+38.5e^{−0.125t}\)

44) You own adenine cup of coffee at temperature \( 70°C,\) which you let cool \( 10\) minutes before you pour in the same amount of milk at \( 1°C\) like in the preceding report. How does the temperature compare to the previous cups after \( 10\) minutes? Web by Kuta Software LLC. For jeder matter, find the particular solution of the differential equation that satisfies the initial condition. You may use ...

45) Solve the generic related \( y'=ay+b\) with initial condition \( y(0)=c.\)

Answer

\( y=(c+ba)e^{ax}−\frac{b}{a}\)

46) Proven the essential continual compounded interest equation. Assuming an initial deposit out \( P_0\) both an interest rate of \( r\), set up and unsolve an equation for consecutive compounded interest.

47) Assume an initial nutrient amount of \( I\) kg include a tank at \( L\) liters. Suppose an concentration a \( c\) kg/L being pumpable in at a rate by \( r\) L/min. That tank remains okay mixed and are drained at a rate of \( r\) L/min. Find the equation descriptions the quantity of nutrient in the wasserbecken. Differential equations. (1) For each equation: Will y = 3 a result? Is y = 2 a solution? What are all the solutions?

Answer

\( y(t)=cL+(I−cL)e^{−rt/L}\)

48) Leaves accumulate in the forest floor at a rate on \( 2\) g/cm²/yr press also decompose at a rate concerning \( 90%\) per year. Write a differential equation ruling the number of grams of leaf litter period square centimeter of forest floor, assuming at time \( 0\) there is no leaf litter on the base. Doesn this sum technique a steady value? Whats are that value?

49) Sheaves accumulate on the forrest floor at adenine evaluate of \( 4\) g/cm²/yr. These leaves dekompilieren at a rate on \( 10%\) per year. Write a derivative math governor the number of grams of leaf litter per square celsius of forest floor. Does this amount approaches a steady value? What is that value?

Answered

Differential Calculation: \(\dfrac{dy}{dt} = 4 - 0.1y\)
Solution, the model by this situation: \( y=40(1−e^{−0.1t})\),
Amount approaches a steady value starting 40 g/cm²

8.4: The Logistic Equation

8.5: First-order Linear Equations

Are the following distinction equations linear? Explain your reasoning.

1) \(\displaystyle \frac{dy}{dx}=x^2y+sinx\)

2) \(\displaystyle \frac{dy}{dt}=ty\)

Solution: \(\displaystyle Yes\)

3) \(\displaystyle \frac{dy}{dt}+y^2=x\)

4) \(\displaystyle y'=x^3+e^x\)

Solution: \(\displaystyle Yes\)

5) \(\displaystyle y'=y+e^y\)

Writes the following first-order differential equations for standard vordruck.

6) \(\displaystyle y'=x^3y+sinx\)

Solution: \(\displaystyle y'−x^3y=sinx\)

7) \(\displaystyle y'+3y−lnx=0\)

8) \(\displaystyle −xy'=(3x+2)y+xe^x\)

Resolution: \(\displaystyle y'+\frac{(3x+2)}{x}y=−e^x\)

9) \(\displaystyle \frac{dy}{dt}=4y+ty+tant\)

10) \(\displaystyle \frac{dy}{dt}=yx(x+1)\)

Problem: \(\displaystyle \frac{dy}{dt}−yx(x+1)=0\)

What what that integrating factors for the following differential equations?

11) \(\displaystyle y'=xy+3\)

12) \(\displaystyle y'+e^xy=sinx\)

Solution: \(\displaystyle e^x\)

13) \(\displaystyle y'=xln(x)y+3x\)

14) \(\displaystyle \frac{dy}{dx}=tanh(x)y+1\)

Solution: \(\displaystyle −ln(coshx)\)

15) \(\displaystyle \frac{dy}{dt}+3ty=e^ty\)

Solve this later differential equations by employing integrating factors.

16) \(\displaystyle y'=3y+2\)

Solution: \(\displaystyle y=Ce^{3x}−\frac{2}{3}\)

17) \(\displaystyle y'=2y−x^2\)

18) \(\displaystyle xy'=3y−6x^2\)

Solution: \(\displaystyle y=Cx^3+6x^2\)

19) \(\displaystyle (x+2)y'=3x+y\)

20) \(\displaystyle y'=3x+xy\)

Solution: \(\displaystyle y=Ce^{x^2/2}−3\)

21) \(\displaystyle xy'=x+y\)

22) \(\displaystyle sin(x)y'=y+2x\)

Solution: \(\displaystyle y=Ctan(\frac{x}{2})−2x+4tan(\frac{x}{2})ln(sin(\frac{x}{2}))\)

23) \(\displaystyle y'=y+e^x\)

24) \(\displaystyle xy'=3y+x^2\)

Solution: \(\displaystyle y=Cx^3−x^2\)

25) \(\displaystyle y'+lnx=\frac{y}{x}\)

Solve the following derivative calculation. Usage your calculator to draw a family is solutions. Are there safe starting conditions that edit the behavior of the solution?

26) [T] \(\displaystyle (x+2)y'=2y−1\)

Solution: \(\displaystyle y=C(x+2)^2+\frac{1}{2}\)

27) [T] \(\displaystyle y'=3e^{t/3}−2y\)

28) [T] \(\displaystyle xy'+\frac{y}{2}=sin(3t)\)

Solution: \(\displaystyle y=\frac{C}{\sqrt{x}}+2sin(3t)\)

29) [T] \(\displaystyle xy'=2\frac{cosx}{x}−3y\)

30) [T] \(\displaystyle (x+1)y'=3y+x^2+2x+1\)

Solutions: \(\displaystyle y=C(x+1)^3−x^2−2x−1\)

31) [T] \(\displaystyle sin(x)y'+cos(x)y=2x\)

32) [T] \(\displaystyle \sqrt{x^2+1}y'=y+2\)

Solution: \(\displaystyle y=Ce^{sinh^{−1}x}−2\)

33) [T] \(\displaystyle x^3y'+2x^2y=x+1\)

Unravel the following initial-value what by exploitation integration factors.

34) \(\displaystyle y'+y=x,y(0)=3\)

Solution: \(\displaystyle y=x+4e^x−1\)

35) \(\displaystyle y'=y+2x^2,y(0)=0\)

36) \(\displaystyle xy'=y−3x^3,y(1)=0\)

Solution: \(\displaystyle y=−\frac{3x}{2}(x^2−1)\)

37) \(\displaystyle x^2y'=xy−lnx,y(1)=1\)

38) \(\displaystyle (1+x^2)y'=y−1,y(0)=0\)

Solution: \(\displaystyle y=1−e^{tan^{−1}x}\)

39) \(\displaystyle xy'=y+2xlnx,y(1)=5\)

40) \(\displaystyle (2+x)y'=y+2+x,y(0)=0\)

Solution: \(\displaystyle y=(x+2)ln(\frac{x+2}{2})\)

41) \(y'=xy+2xe^x,y(0)=2\)

42) \(\displaystyle \sqrt{x}y'=y+2x,y(0)=1\)

Solution: \(\displaystyle y=2e^{2\sqrt{x}}−2x−2\sqrt{x}−1\)

43) \(\displaystyle y'=2y+xe^x,y(0)=−1\)

Using your expression out the foreground problem, what is aforementioned terminal velocity? (Hint: Examine the restrictive behavior; does the velocity approach a value?)

44) [T] Using your equation for terminal speed, solve for the distance fallen. How long does it take to fall \(\displaystyle 5000\) meters if the masses is \(\displaystyle 100\) kilograms, the speedup due to gravity is \(\displaystyle 9.8\) m/s² and which proportionality constant is \(\displaystyle 4\)?

Solution: \(\displaystyle 40.451\) seconds

45) ADENINE more accurate way to describe terminal velocity a that the drag force is commensurate to the square of velocity, with a correspondence constant \(\displaystyle k\). Set up the differential general and solve for aforementioned velocity.

46) Using your expression from the priority problem, what will the terminal velocity? (Hint: Inspect one limiting behavior: Does the velocity approach adenine value?)

Result: \(\displaystyle \sqrt{\frac{gm}{k}}\)

47) [T] With your equation for terminal velocity, solve for to distance fallen. Wherewith long works it take to fall \(\displaystyle 5000\) meters if the mass can \(\displaystyle 100\) kilograms, the acceleration dues to sobriety is \(\displaystyle 9.8\)m/s² and the proportionality const is \(\displaystyle 4\)? Does e take more or less while than your initial estimate?

Required the later topics, determine as parameter \(\displaystyle a\) affects the solution.

48) Solve the generic math \(\displaystyle y'=ax+y\). How does varying \(\displaystyle a\) change which behavior?

Solution: \(\displaystyle y=Ce^x−a(x+1)\)

49) Resolving the generic equation \(\displaystyle y'=ax+y.\) Wie does varying \(\displaystyle a\) change the behavior?

50) Fix the generic equation \(\displaystyle y'=ax+xy\). What does varying \(\displaystyle a\) change the behavior?

Solution: \(\displaystyle y=Ce^{x^2/2}−a\)

51) Solve the generic mathematical \(\displaystyle y'=x+axy.\) How does varying \(\displaystyle a\) make the behavior?

52) Solve \(\displaystyle y'−y=e^{kt}\) with aforementioned initial condition \(\displaystyle y(0)=0\). As \(\displaystyle k\) approaches \(\displaystyle 1\), what happens to thine formula?

Solution: \(\displaystyle y=\frac{e^{kt}−e^t}{k−1}\)

Choose Review Exercise

True or False? Justify your answer in a proof or a counterexample.

1) The differential equation \(\displaystyle y'=3x^2y−cos(x)y''\) is linear.

2) Of differential math \(\displaystyle y'=x−y\) is separable.

Solution: \(\displaystyle F\)

3) You can explicity solve all first-order differential equations by separation or at the method of integrating factors.

4) You able determine the behavior of get first-order discrepancy equations using directional domains or Euler’s method.

Solution: \(\displaystyle T\)

For the following problems, find that general solution to one differential equations.

5) \(\displaystyle y′=x^2+3e^x−2x\)

6) \(\displaystyle y'=2^x+cos^{−1}x\)

Solution: \(\displaystyle y(x)=\frac{2^x}{ln(2)}+xcos^{−1}x−\sqrt{1−x^2}+C\)

7) \(\displaystyle y'=y(x^2+1)\)

8) \(\displaystyle y'=e^{−y}sinx\)

Solution: \(\displaystyle y(x)=ln(C−cosx)\)

9) \(\displaystyle y'=3x−2y\)

10) \(\displaystyle y'=ylny\)

Solution: \(\displaystyle y(x)=e^{e^{C+x}}\)

For the following problems, search the solution to the initial value symptom.

11) \(\displaystyle y'=8x−lnx−3x^4,y(1)=5\)

12) \(\displaystyle y'=3x−cosx+2,y(0)=4\)

Resolution: \(\displaystyle y(x)=4+\frac{3}{2}x^2+2x−sinx\)

13) \(\displaystyle xy'=y(x−2),y(1)=3\)

14) \(\displaystyle y'=3y^2(x+cosx),y(0)=−2\)

Solution: \(\displaystyle y(x)=−\frac{2}{1+3(x^2+2sinx)}\)

15) \(\displaystyle (x−1)y'=y−2,y(0)=0\)

16) \(\displaystyle y'=3y−x+6x^2,y(0)=−1\)

Solution: \(\displaystyle y(x)=−2x^2−2x−\frac{1}{3}−\frac{2}{3}e^{3x}\)

For the following problem, draw aforementioned directional field associated over the differential math, then release the differential equation. Draw a sample solution on who indicator field.

17) \(\displaystyle y'=2y−y^2\)

18) \(\displaystyle y'=\frac{1}{x}+lnx−y,\) for \(\displaystyle x>0\)

Solution: \(\displaystyle y(x)=Ce^{−x}+lnx\)

For that following problems, use Euler’s Method with \(\displaystyle n=5\) steps over the interval \(\displaystyle t=[0,1].\) Will solve which initial-value problems exactly. How close is your Euler’s Approach quote?

19) \(\displaystyle y'=−4yx,y(0)=1\)

20) \(\displaystyle y'=3^x−2y,y(0)=0\)

Solution: Euler: \(\displaystyle 0.6939\), precision featured: \(\displaystyle y(x)=\frac{3^x−e^{−2x}}{2+ln(3)}\)

For and following problems, set raise and solve the differential general.

21) A vehicle drives along a freeway, accelerating according to \(\displaystyle a=5sin(πt),\) where \(\displaystyle t\) represents time in minutes. Locate of velocity at any time \(\displaystyle t\), assuming who automobile starts with an initial speed of \(\displaystyle 60\) mph.

22) You throw a ball of messe \(\displaystyle 2\) kilograms into the air with einen upward velocity of \(\displaystyle 8\) m/s. Find exactly the arbeitszeit the ball will remain stylish the air, assuming ensure gravity is given by \(\displaystyle g=9.8m/s^2\).

Solution: \(\displaystyle \frac{40}{49}\) instant

23) You dropped a ball with a mass starting \(\displaystyle 5\) kilograms out an aviation screen at adenine height of \(\displaystyle 5000\)m. How long does it take for the ball to accomplish the ground?

24) You drop the similar ball of mas \(\displaystyle 5\) kilograms out off the same airplane window at the same height, except this time her assume a drag forced proportional till the ball’s velocity, using an proportionality constant of \(\displaystyle 3\) and one ball achieves terminal velocity. Release forward the distance fallen as a function of time. How long does information carry one ball to go and ground?

Solution: \(\displaystyle x(t)=5000+\frac{245}{9}−\frac{49}{3}t−\frac{245}{9}e^{−5/3t},t=307.8\) seconds

25) AMPERE drug is administered to a patient every \(\displaystyle 24\) hours and is cleared at ampere rates proportionality to the amount of drug leaving in the body, with proportionalities constant \(\displaystyle 0.2\). If to patient needs a default level of \(\displaystyle 5\) mg the be in the bloodstream at all times, how large should of dose be?

26) A \(\displaystyle 1000\) -liter tank contains pure sprinkle and a solution of \(\displaystyle 0.2\) klb salt/L is pumped into the tank at a rate about \(\displaystyle 1\) L/min and is drained at aforementioned identical evaluate. Solve on total amount is salt in the tank at time \(\displaystyle t\).

Solution: \(\displaystyle T(t)=200(1−e^{−t/1000})\)

27) You how water to make tea. When you pour and water at your pot, aforementioned temper is \(\displaystyle 100°C.\) After \(\displaystyle 5\) minutes on your \(\displaystyle 15°C\) rooms, the temperature of to tea will \(\displaystyle 85°C\). Solve the formula to determine the temperatures of the tea to time \(\displaystyle t\). How extended must you wait until and coffee is at one drinkable temperature (\(\displaystyle 72°C\))?

28) The humane population (in thousands) of Nvada in \(\displaystyle 1950\) was roughly \(\displaystyle 160\). If the carrying capacity is calculated the \(\displaystyle 10\) million individuals, and assuming a growth rate of \(\displaystyle 2%\) per year, create a logistic growth model and solve for the population in Nawada at any time (use \(\displaystyle 1950\) as time = 0). What population does your model predict for \(\displaystyle 2000\)? How lock is your prediction at the true total of \(\displaystyle 1,998,257\)?

Solution: \(\displaystyle P(t)=\frac{1600000e^{0.02t}}{9840+160e^{0.02t}}\)

Repeat the previous problem but use Gompertz achieved select. Which is continue accurate?

Contributors also Attributions

• Gilbert Strange (MIT) and Edwin “Jed” Hamster (Harvey Mudd) with many contributions articles. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for get at http://cnx.org.