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# Calculus exercises: part IV

## Contents

## Basics

$\bullet$ Find the eigenvalues and the eigenvectors of the following matrix: $$F=\left[\begin{array}{c}2&1\\1&2\end{array}\right].$$

$\bullet$ Verify that the function $y=cx^{2}$ is a solution of the differential equation: $$xy'=2y.$$ Are there any others?

$\bullet$ Find all curves perpendicular to the family of curves: $$xy^{2}=C.$$

$\bullet$ Find all curves perpendicular to the family of curves: $$x^{2}y=C.$$

$\bullet$ Verify that the function $y=cx^{2}$ is a solution of the differential equation: $$xy'=2y.$$ Are there any others?

$\bullet$ Find all curves perpendicular to the family of curves: $$xy^{2}=C.$$

## Analytical methods

$\bullet$ Solve by separating variables: $$y'=xy.$$

$\bullet$ Solve by the method of integrating factor: $$y'=y/x.$$

$\bullet$ Solve the differential equation: $$y'+2xy^2=0.$$

$\bullet$ Solve the differential equation: $$x\frac{dy}{dx}-y=x^2\sin x.$$

$\bullet$ Solve the IVP: $$(4y+2x-5)\, dx+(6y+4x-1)\, dy=0,\ y(-1)=2.$$

$\bullet$ Solve the differential equation by using an appropriate substitution: $$\frac{dy}{dx}=1+e^{y-x+5}.$$

$\bullet$ Solve the differential equation by using an appropriate substitution: $$\frac{dy}{dx}=2+\sqrt{y-2x+3}.$$

$\bullet$ Suppose $y$ is the solution of the initial value problem $$y'=x^2+y^2,\ y(0)=0.$$ Find $y(1)$ by means of Euler's method with step $h=.2$.

$\bullet$ Solve the differential equation $$xyy'=x^2+3y^2.$$

$\bullet$ Solve the differential equation $$(2x^2y+3x^2)\, dx+(2x^2y+4y^3)\, dy.$$

$\bullet$ Solve the differential equation: $$y'=-6xy.$$

## Euler's method

$\bullet$ Carry out $n=4$ steps of Euler's method with $h=.5$ for the following initial value problem: $$y'=y-x,\ y(0)=2.$$

$\bullet$ Use Euler's method with $n=4$ steps to estimate the solution of the initial values problem: $$y'=2x-y,\ y(0)=1,$$ on the interval $[0,1]$.

$\bullet$ Use Euler's method with $4$ steps to estimate the solution of the initial values problem: $$y'=2x-y,\ y(0)=1,$$ on the interval $[0,1]$.

## Generalities

$\bullet$ Provide the definition of the uniqueness property and sketch an example of a solution set without it.

$\bullet$ Indicate if the following statements are true or false.

- $y'=y^2$ is a second order ODE.
- $y(t)=|t|$ is a solution of the IVP $y'=1, y(0)=0$.
- $y(t)=-t$ is a solution of a DE with the slope field below.
- The IVP $y′=\frac{1}{t^2+1},\ y(0)=1$ has more than one solution.
- The ODE $y′=yt^2\sin t$ can be solved by separation of variables.

$\bullet$ Verify that $-2x^{2}y+y^{2}=1$ is an implicit solution of the differential equation $$2xy+(x^2-y)\frac{dy}{dx}=0.$$ Find one explicit solution.

$\bullet$ (a) State the Existence-Uniqueness Theorem for a system of two linear equations. (b) For what values $t_0,a,b$ does the theorem guarantee existence and uniqueness of the IVP $$x'=y/t,\ y'=x,\ x(t_0)=a,\ y(t_0)=b.$$

## Models and setting up ODEs

$\bullet$ What is the ODE of an object moving horizontally through a medium whose resistance is proportional to the object's velocity? Describe the long term behavior of the object.

$\bullet$ (a) In the theory of learning, the rate at which a subject is memorized is assumed proportional to the amount that is left to be memorized. Suppose $M$ denotes the total amount of subject to be memorized and $A(t)$ is the amount memorized at time $t$. Set up a differential equation for $A(t)$. (b) Suppose in addition that the rate at which material is forgotten is proportional to $A(t)$. Set up a differential equation for $A(t)$.

$\bullet$ Suppose point $T$ goes along the line $x=1$ while dragging point $P$ on the $xy$-plane by a string $PT$ of length $1$. Suppose $T$ starts at $(1,0)$ and $P$ at $(2,0)$. Find the path of $P$.

$\bullet$ Set up (don't solve) initial values problems for the following two situations: (a) An object is thrown up from a building of height $h$ at $45$ degrees with speed $s$; (b) An object thrown travels for 2 seconds and then hits the ground at $45$ degrees and speed $s.$.

$\bullet$ Set up and solve the differential equation that describes the motion of an object of mass $M$ placed on top of a spring with Hooke constant $k$ standing vertically on the ground.

$\bullet$ (a) Describe the predator-prey model. (b) Set up a system of differential equations for the model and find its equilibria.

$\bullet$ Suppose an object is moving horizontally through a medium whose resistance is proportional to the object's velocity: $$v'=-ky,\ k>0.$$ Describe the long term behavior of the object. What if $k$ grows with time?

$\bullet$ Derive the equations describing the predator-prey system and plot the phase portrait.

$\bullet$ Set up and solve the differential equation that describes the motion of an object of mass $M$ suspended vertically by a spring with Hooke constant $k$.

$\bullet$ Set up (don't solve) initial values problems for the following two situations: (a) An object is thrown up from a building of height $h$ at $45$ degrees with speed $s$; (b) An object thrown travels for $2$ seconds and then hits the ground at $4$5 degrees and speed $s$.

$\bullet$ Set up and solve the differential equation that describes the motion of an object of mass $M$ placed on top of a spring with Hooke constant $k$ standing vertically on the ground.

$\bullet$ (a) Describe the predator-prey model. (b) Set up a system of differential equations for the model and find its equilibria.

## Qualitative analysis

$\bullet$ A sketch of the solution set of an ODE $y'=f(t,y)$ is shown below. What can you say about the sign of $f$?

$\bullet$ Suppose $y=0$ is a stable (unstable) equilibrium of $y'=f(x,y)$. Suppose $g(x,y)=f(x,y)$ if $|y|<1/x$. What can you tell about $y'=g(x,y)$?

$\bullet$ In the square $[-3,3]\times[-3,3]$, plot the direction field for the differential equation $$y\frac{dy}{dx}=-x.$$ Sketch the stationary solution and three other solution curves.

$\bullet$ Sketch the direction field of the system: $$x'=2x-y,\ y'=x-3y$$ and several of its trajectories, identify the type of equilibrium.

$\bullet$ According to Newton's law of cooling, the change of the temperature $T$ of a body immersed in a medium of constant temperature $A$ is described by the differential equation, with respect to time $x$: $$\frac{dT}{dx}=k(A-T),\ k>0.$$ Solve it. Interpret your solution to explain why the temperature of the body will become equal to $A$, eventually.

## Systems

$\bullet$ Sketch the trajectories of a system of linear ODEs $X'=FX$ if the matrix $F$ has these pairs of eigenvalues and eigenvectors: $$\lambda_1=2,\ V_1=\left[\begin{array}{c}1\\0\end{array}\right] \text{ and } \lambda_2=-1,\ V_2=\left[\begin{array}{c}1\\1\end{array}\right].$$

$\bullet$ Write the general solution of a system of linear ODEs $X'=FX$ if the matrix $F$ has these pairs of eigenvalues and eigenvectors:
$$\lambda_1=i,\ V_1=\left[\begin{array}{c}1\\i\end{array}\right] \text{ and } \lambda_2=-i,\ V_2=\left[\begin{array}{c}1\\-i\end{array}\right],$$
and provide one non-trivial *real* solution of the system.

$\bullet$ Solve the system of differential equations: $$x'=2x-y,\ y'=x.$$

$\bullet$ Solve the system of linear equations: $$x'=4x-5y,\ y'=5x-4y.$$

$\bullet$ Suppose $\lambda_1,\ \lambda_2$ are the eigenvalues of a system of linear ordinary differential equations. Sketch the phase portrait of the system if (a) $\lambda_1,\ \lambda_2$ are complex with $\operatorname{Re} \lambda_1<0$; (b) $\lambda_1=\lambda_2$ with a single linearly independent eigenvector; (c) $\lambda_1,\ \lambda_2$ are real of opposite signs.

$\bullet$ Solve the system $$x'=-3x-4y,\ y'=2x+y.$$

## Second order

$\bullet$ Find the general solutions to these differential equations with constant coefficients:

- (1) $y' ' -6y+9y=0$,
- (2) $2y' '-5y'-3y=0$,
- (3) $y' ' +4y'+7y=0$.

$\bullet$ The differential equation $xy' '+y'=0$ has a known solution $y_1=\ln x$. Find another solution $y_2$ linearly independent from the first.

$\bullet$ The function $y_1=e^x$ is a solution of the homogeneous equation $$y' '-3y'+2y=0.$$ Solve the non-homogeneous equation $$y' '-3y+2y=5e^{3x}.$$

$\bullet$ Given the differential equation $y' '-2y+2y=0$, solve (a) the initial value problem $$y(\pi )=1,\ y'(\pi )=1,$$ and (b) the boundary value problem $$y(0)=1,\ y(\pi )=1.$$

$\bullet$ Solve the differential equation: $$x^2y' ' +(y')^2=0.$$

$\bullet$ Solve the initial value problem $$(x-1)y' '-xy+y=0,\ y(0)=-2,\ y'(0)=6,$$ in the form of a power series. (One extra point for representing the solution in a more compact form.)

$\bullet$ Solve the following differential equation: $$y' '+2y'+4y=0.$$

$\bullet$ Solve the differential equation $$y' '+y'x=0.$$

$\bullet$ Solve the following initial value problem: $$y' '+2y=0,\ y(0)=0,\ y'(0)=1.$$

$\bullet$ Solve the following differential equation: $$y' '+2y'+4y=0.$$

## Advanced

$\bullet$ Solve the boundary value problem: $$y' '-4y=0,\ y(0)=1,\ y(1)=2.$$

$\bullet$ Verify that the power series $$y=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}x^n$$ is a solution of the differential equation $$(x+1)y' ' +y'=0.$$

$\bullet$ Given a differential equation $$(\cos x)y' '+y=0,$$ find two linearly independent solutions in the form of power series. Provide all terms up to $x^2$. Recall that $$\cos x=1-\frac{1}{2}x^2+\frac{1}{4!}x^4-...$$

$\bullet$ Provide the power series solution for $$y'+y=1.$$

$\bullet$ Suppose $y_1,\ y_2$ are two solutions of the equation $$y' ' +p(x)y'+q(x)=0$$ and $c_1,c_2$ are constants. Show that $y=c_1y_1+c_2y_2$ is also a solution of the equation.

$\bullet$ Find a general form of a particular solution of $$y^{(3)}+y' '=3e^x+4x^2.$$

$\bullet$ By the power series method, solve the IVP $$y' '+y'-2y=0,\ y(0)=1,\ y'(0)=-2.$$

$\bullet$ Solve the following homogeneous equation: $$2xy\frac{dy}{dx}=4x^{2}+3y^{2}.$$

$\bullet$ Solve this exact equation $$y^{3}\,dx+3y^{2}x\,dy=0.$$

$\bullet$ Solve the following differential equation in the complex domain: $$y'+y'+y=0.$$

$\bullet$ Provide the power series solution for $$y'+y=1.$$

## PDEs

## Computing

*Computing assignment #1: Free fall*. Carry out, or finish, or reproduce one of these projects from calculus 1 (the output should be a short presentation demonstrating an Excel spreadsheet accompanied by an explanation):

- 1. How do I throw a ball down a staircase so that it bounces off each step? (fall 2016)
- 2. How should you throw a ball from the top of a $100$ story building so that it hits the ground at $100$ feet per second? (fall 2016)
- 3. I would like to use a cannon with a muzzle velocity of $100$ feet per second to bombard the inside of a fortification $300$ feet away with walls $20$ feet high. (fall 2016)
- 4. I have a toy cannon and I want to shoot it from a table and hit a spot on the floor $10$ feet away from the table. (fall 2016)
- 5. How hard do I have to push a toy truck from the floor up a $30$ degree incline to make it reach the top of the table at zero speed? (fall 2016)
- 6. How fast does the shadow of a falling ball on a sliding ladder move?
- 7. How fast do I have to move my hand while spinning a sling in order to throw the rock $100$ feet away? (fall 2016)

*Computing assignment #2: Euler's method*. For one of the ODEs below, subject it to Euler's method analysis with Excel:

- Analyze the domain of the right-hand side, apply the existence and uniqueness theorems.
- Plot sufficiently many solutions.
- Find patterns of the set of solutions, such as: periodicity, monotonicity, asymptotes, symmetry, and any other.

ODEs:

- 1. $y'=\tan (y^2)$;
- 2. $y'=\cos (x+y)$;
- 3. $y'=\cos y \cdot \sqrt{|y|}$;
- 4. $y'=\sin y \cdot \sqrt{ y}$;
- 5. $y'=[x+y]$ (the FLOOR function in Excel);
- 6. $y'=\frac{1}{x-y}$.

*Computing assignment #3: Euler's method on the plane*. For one of the systems of ODEs below, perform the same tasks as in the last assignment:

- 1. $x'=\sin y,\ y'=\tan (y^2)$;
- 2. $x'=|y|,\ y'=\cos (x+y)$;
- 3. $x'=x+y,\ y'=\cos y \cdot \sqrt{|y|}$;
- 4. $x'=y,\ y'=\sin y \cdot \sqrt{ y}$;
- 5. $x'=x-y,\ y'=[x+y]$ (the FLOOR function in Excel);
- 6. $x'=\cos(x+y),\ y'=\frac{1}{x-y}$.

*Computing assignment #4: Van der Pol oscillator*. Apply the qualitative methods and Euler's method to analyze the following second order equation:
$$x' '=m(1-x^2)x'-x.$$

## Random

Sample exams from 2006: