12.5: Problems

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Exercise $$\PageIndex{1}$$

Find all of the solutions of the first order differential equations. When an initial condition is given, find the particular solution satisfying that condition.

1. $$\frac{d y}{d x}=\frac{e^{x}}{2 y}$$.
2. $$\frac{d y}{d t}=y^{2}\left(1+t^{2}\right), y(0)=1$$.
3. $$\frac{d y}{d x}=\frac{\sqrt{1-y^{2}}}{x}$$.
4. $$x y^{\prime}=y(1-2 y), \quad y(1)=2$$.
5. $$y^{\prime}-(\sin x) y=\sin x$$.
6. $$x y^{\prime}-2 y=x^{2}, y(1)=1$$.
7. $$\frac{d s}{d t}+2 s=s t^{2}, \quad, s(0)=1 .$$
8. $$x^{\prime}-2 x=t e^{2 t}$$.
9. $$\frac{d y}{d x}+y=\sin x, y(0)=0$$.
10. $$\frac{d y}{d x}-\frac{3}{x} y=x^{3}, y(1)=4$$.

Exercise $$\PageIndex{2}$$

Consider the differential equation $\frac{d y}{d x}=\frac{x}{y}-\frac{x}{1+y} .\nonumber$

1. Find the 1-parameter family of solutions (general solution) of this equation.
2. Find the solution of this equation satisfying the initial condition $$y(0)=1$$. Is this a member of the 1-parameter family?

Exercise $$\PageIndex{3}$$

Identify the type of differential equation. Find the general solution and plot several particular solutions. Also, find the singular solution if one exists.

1. $$y=x y^{\prime}+\frac{1}{y^{\prime}}$$.
2. $$y=2 x y^{\prime}+\ln y^{\prime}$$.
3. $$y^{\prime}+2 x y=2 x y^{2}$$.
4. $$y^{\prime}+2 x y=y^{2} e^{x^{2}}$$.

Exercise $$\PageIndex{4}$$

Find all of the solutions of the second order differential equations. When an initial condition is given, find the particular solution satisfying that condition.

1. $$y^{\prime \prime}-9 y^{\prime}+20 y=0$$.
2. $$y^{\prime \prime}-3 y^{\prime}+4 y=0, \quad y(0)=0, \quad y^{\prime}(0)=1$$.
3. $$8 y^{\prime \prime}+4 y^{\prime}+y=0, \quad y(0)=1, \quad y^{\prime}(0)=0$$.
4. $$x^{\prime \prime}-x^{\prime}-6 x=0$$ for $$x=x(t)$$.

Exercise $$\PageIndex{5}$$

Verify that the given function is a solution and use Reduction of Order to find a second linearly independent solution.

1. $$x^{2} y^{\prime \prime}-2 x y^{\prime}-4 y=0, \quad y_{1}(x)=x^{4}$$.
2. $$x y^{\prime \prime}-y^{\prime}+4 x^{3} y=0, \quad y_{1}(x)=\sin \left(x^{2}\right)$$.

Exercise $$\PageIndex{6}$$

Prove that $$y_{1}(x)=\sinh x$$ and $$y_{2}(x)=3 \sinh x-2 \cosh x$$ are linearly independent solutions of $$y^{\prime \prime}-y=0$$. Write $$y_{3}(x)=\cosh x$$ as a linear combination of $$y_{1}$$ and $$y_{2}$$.

Exercise $$\PageIndex{7}$$

Consider the nonhomogeneous differential equation $$x^{\prime \prime}-3 x^{\prime}+2 x=6 e^{3 t}$$.

1. Find the general solution of the homogenous equation.
2. Find a particular solution using the Method of Undetermined Coefficients by guessing $$x_{p}(t)=A e^{3 t}$$.
3. Use your answers in the previous parts to write down the general solution for this problem.

Exercise $$\PageIndex{8}$$

Find the general solution of the given equation by the method given.

1. $$y^{\prime \prime}-3 y^{\prime}+2 y=10$$. Method of Undetermined Coefficients.
2. $$y^{\prime \prime}+y^{\prime}=3 x^{2}$$. Variation of Parameters.

Exercise $$\PageIndex{9}$$

Use the Method of Variation of Parameters to determine the general solution for the following problems.

1. $$y^{\prime \prime}+y=\tan x$$.
2. $$y^{\prime \prime}-4 y^{\prime}+4 y=6 x e^{2 x}$$.

Exercise $$\PageIndex{10}$$

Instead of assuming that $$c_{1}^{\prime} y_{1}+c_{2}^{\prime} y_{2}=0$$ in the derivation of the solution using Variation of Parameters, assume that $$c_{1}^{\prime} y_{1}+c_{2}^{\prime} y_{2}=h(x)$$ for an arbitrary function $$h(x)$$ and show that one gets the same particular solution.

Exercise $$\PageIndex{11}$$

Find all of the solutions of the second order differential equations for $$x>0$$.. When an initial condition is given, find the particular solution satisfying that condition.

1. $$x^{2} y^{\prime \prime}+3 x y^{\prime}+2 y=0$$.
2. $$x^{2} y^{\prime \prime}-3 x y^{\prime}+3 y=0$$.
3. $$x^{2} y^{\prime \prime}+5 x y^{\prime}+4 y=0$$.
4. $$x^{2} y^{\prime \prime}-2 x y^{\prime}+3 y=0$$.
5. $$x^{2} y^{\prime \prime}+3 x y^{\prime}-3 y=x^{2}$$.

This page titled 12.5: Problems is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Russell Herman via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.