12.5: Problems

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. \(x^{2} y^{\prime \prime}+3 x y^{\prime}-3 y=x^{2}\).