## Homework

Unit 1

Assignment 1

1. Determine whether the given function is a solution to the differential equation:
1. $y=\frac{x^3}{3}+\pi$   ,   $\frac{dy}{dx}=x^2$
2. $y=\frac{1}{(2-x)^2}$   ,   $y'=y^2$
3. $y=\sqrt{4-x^2}$   ,   $y\frac{dy}{dx}+x=0$
4. $x=\cos{\left(\sqrt{\frac{k}{m}}\cdot t\right)}$   ,   $m\ddot{x}+kx=0$
5. $y=e^{-x}+e^x$   ,   $y''=y$
6. $y=ax^2+bx+c$   ,   $\frac{d^3y}{dx^3}=0$
2. Use implicit differentiation to show that $x^2+y^2=4$ is a solution to $\frac{dy}{dx}=-\frac{x}{y}$.
3. For which values of $m$ is $\phi(x)=e^{mx}$ a solution to $\frac{d^2y}{dx^2}+6\frac{dy}{dx}+5y=0$
4. Solve the following differential equations by integrating or separation of variables:
1. $y'=x^2$  ,    $y(-3)=2$
2. $\frac{dy}{dx}=\frac{y-1}{x+3}$  ,    $(-1,0)$
3. $x^2dx=-ydy$
4. $\sin(2\theta)d\theta+\cos(y)dy=0$
5. $\frac{dy}{dx}=\frac{4xy}{x^2+1}$
6. $y''=1$  ,    $y(0)=1$  ,  $y'(0)=2$

Assignment 2

1. Draw a slope field for $\frac{dy}{dx}=y(2-y)$  for $0\le x\le 4$ and $0\le y\le 4$.  Draw the slope segments every 0.25 units.  Then, draw a couple of curves through the field.
2. Use a slope field application to create a slope field for $\frac{dy}{dx}=x-y$.  Use Euler's method to estimate a solution curve with the initial value, $y(0)=1$ and $\delta x = 0.2$.
3. Use a slope field application to create a slope field for $\frac{dy}{dx}=p(p-1)(2-p)$.  Sketch curves for the initial values $y=4,\; y=1.5,$ and $y=0.5$, when $x=0$.

Assignment 3

1. $y'-\frac{1}{x}y=x^3$
2. $\frac{dy}{dx}-\frac{2y}{x}=x^5$ , $x>0$
3. $y'-2xy=1$  ,    $y(0)=1$
4. $y'-3x=0$
5. $y'-3x=15$
6. $y'+y=0$  ,   $y(-1)=3$
7. $y'+y=e^-x$  ,  $y(-1)=3$
8. $y'+y\sin(x)=0$  ,  $y(0)=0$
9. $y'+y\sin(x)=kx$  ,   $y(0)=0$
10. Rewrite $p(x)=a+\int_0^r t^2 p(t) dt$ as a differential equation and solve for $p(r)$
11. $\cos(x)\frac{dy}{dx}+y\,\sin(x)=2x\,\cos^2(x)$  ,    $y\left(\frac{\pi}{4}\right)=\frac{-15\sqrt{2}\pi^2}{32}$
12. Bernoulli Equations: $\frac{dy}{dx}+2y=xy^{-2}$ .  Substitute $v=y^3$ to convert to a linear differential equation.
13. A tank is initially filled with 1000 L of water with 6 kg of salt. A brine containing 0.3 kg of salt per liter runs into the tank at 5 L/min.  The mixture is kept uniformly mixed and is flowing out at 6 L/min.  How much salt is in the tank after 8 minutes?

Assignment 4

1. Solve: $(x+y^2)dx+(y-x^2)dy=0$
2. Solve: $(3x^2y^2+x^2)dx+(2x^3y+y^2)dy=0$
3. Solve: $(2x+ye^{xy})dx+(xe^{xy})dy=0$
4. Solve: $(3x^2+2xy^2)dx+(2x^2y)dy=0$ where $y(2)=-3$
5. If $y^2\sin(x)dx+yf(x)dy=0$ is an exact equation, then what could $f(x)$ be?

Assignment 5

1. Consider the linear differential equations $\frac{dv}{dt}+\frac{k}{m}v=g$ ,  where $v$ is the velocity of a falling object, $g$ is the acceleration due to gravity, $m$ is the mass of the object, and $k$ is the friction constant due to air resistance.
Solve the initial value problem, where $v(0)=v_o$ and find the terminal velocity, $v_l$
2. With the solution to problem 1 above for $v$, solve the displacement integral for $x$:

3. A body at a temperature of $100^\circ$ is placed in a room of constant temperature.  If after 10 minutes the body has cooled to $90^\circ$ and after 20 minutes to $85^\circ$, then find the temperature of the room.
4. More coming soon . . .

UNIT 2

Assignment 1

1. What is the period of a non-zero solution to $\ddot{x}+4x=0$?
2. Check that $x_1=\cos(\omega t)$ and $x_2=\sin(\omega t)$ are solutions to:

3. Check that $A\cos(\omega t+\phi)$ is also a solution to:

4.  For what values of $r$ is $x=e^{rt}$ a solution to:

5. Find the general solution for $y''+4y'+5y=0$.
6. Find the general solution for $y''+2y'-3y=0$.  Then find a specific solution for the initial conditions $y(0)=1$ and $y'(0)=-1$
7. Find the general solution for $\ddot{x}+2\dot{x}+2x=0$.
8. Find the general solution for $y''+2y'+5y=0$.  Then find a specific solution for the initial conditions $y(0)=1$ and $y'(0)=-1$
9. Find the general solution for $y''+ 6y'+ 9y = 0$.

### 6 Responses to “Homework”

1. andrew says:

Is part 6 of question 4 the initial values for parts 3, 4, and 5?

Thanks!

Andrew

• gary says:

No, The differential equation is y''=1.
Since it is a second order equation, two conditions are needed to find a specific solution. Here the initial conditions are y(0)=1 and y'(0)=2. So, at the point (0,1) the solution curve has a slope of 2.

2. andrew says:

What are the p's in part three of assigment 2?

3. andrew says:

For parts 2 and 3 of assignment 2, are we just supposed to do it on the computer, actually sketch it out on paper, or something else?

Andrew

• Gary says:

If you are going to turn it in for a grade, then you could print it out, or copy it into a file, or email it, or copy it by hand. Otherwise, just look at it.

Some of the applications that I posted also sketch Euler's method for estimating solutions.