Differential equations: exercises

• Indicate if the following statements are true or false.

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

• 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 $y'=g(x,y)$?

• 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.

• (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 $A(t)$.

• 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.

• Solve the differential equation:

$$y'+2xy^2=0.$$

• Solve the differential equation:

$$x\frac{dy}{dx}-y=x^2\sin x.$$

• Solve the IVP:

$$(4y+2x-5)dx+(6y+4x-1)dy=0,\ y(-1)=2.$$

• Solve the differential equation by using an appropriate substitution:

$$\frac{dy}{dx}=1+e^{y-x+5}.$$

• Solve the boundary value problem:

$$y' '-4y=0,\ y(0)=1,\ y(1)=2.$$

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

• Find the general solutions to these differential equations with constant coefficients:
  • (1) $y' ' -6y+9y=0$,
  • (2) $2y' '-5y'-3y=0$,
  • (3) $y' ' +4y'+7y=0$.

• Solve the system of differential equations:

$$x'=2x-y,\ y'=x.$$

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

• 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

$$\cos x=1-\frac{1}{2}x^2+\frac{1}{4!}x^4-...$$

• Solve the differential equation by using an appropriate substitution:

$$\frac{dy}{dx}=2+\sqrt{y-2x+3}.$$

• 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 $h=.2$.

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

• 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 $$y(0)=1,\ y(\pi )=1.$$

• Solve the differential equation:

$$x^2y' ' +(y')^2=0.$$

• 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.)

• Solve the system of linear equations:

$$x'=4x-5y,\ y'=5x-4y.$$

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

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

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

• 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 $P$.

• Solve the following differential equation:

$$y' '+2y'+4y=0.$$

• Provide the power series solution for

$$y'+y=1.$$

• 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 $[0,1]$.

• 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 $s.$.

• 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.

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

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• Solve the differential equation

$$xyy'=x^2+3y^2.$$

• • Solve the differential equation

$$(2x^2y+3x^2)dx+(2x^2y+4y^3)dy.$$

• 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.

• 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.

• Find a general form of a particular solution of

$$y^{(3)}+y' '=3e^x+4x^2.$$

• (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

$$x'=y/t,\ y'=x,\ x(t_0)=a,\ y(t_0)=b.$$

• Derive the equations describing the predator-prey system and plot the phase portrait.

• Solve the system

$$x'=-3x-4y,\ y'=2x+y.$$

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

• By the power series method, solve the IVP

$$y' '+y'-2y=0,\ y(0)=1,\ y'(0)=-2.$$