Chapter 8: Ordinary Differential Equations

Find the general solution of the following differential equations (complementary function + particular solution). Find the particular solution by inspection or by $(\mathbf{6}.\mathbf{18}),(\mathbf{6}.\mathbf{23}),$or.$(\mathbf{6}.\mathbf{24}),$ Alsofind a computer solution and reconcile differences if necessary, noticing especially whether the particular solution is in simplest form [see$\mathbf{(}\mathbf{6}\mathbf{.26}\mathbf{)}$and the discussionafter$\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{15}\mathbf{)}$].

$\mathbf{(}{\mathbf{D}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{8}\mathit{x}\mathit{s}\mathit{i}\mathit{n}\mathit{x}$

For the following problems, verify the given solution and then, by method (e) above, find a second solution of the given equation.

$\left(x^2+1\right)y^{\text{'}\text{'}}-2x{y}^{\text{'}}+2y=0$

Use L28 and L4 to find the inverse transform of$\mathit{p}{\mathit{e}}^{\mathbf{-}\mathbf{p}\mathbf{t}}\mathbf{}\mathit{I}\left(p^2+1\right)$.

Solve the following sets of equations by the Laplace transform method

$$\begin{gathered} \mathit{y}\mathbf{\text{'}}\mathbf{+}\mathit{z}\mathbf{\text{'}}\mathbf{-}\mathbf{3}\mathit{z}\mathbf{=}\mathbf{0}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathit{y}}_{\mathbf{0}}\mathbf{=}\mathit{y}{\mathbf{\text{'}}}_{\mathbf{0}}\mathbf{=}\mathbf{0} \\ \mathit{y}\mathbf{"}\mathbf{+}\mathit{z}\mathbf{\text{'}}\mathbf{=}\mathbf{0}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathit{z}}_{\mathbf{0}}\mathbf{=}\frac{\mathbf{4}}{\mathbf{3}} \end{gathered}$$

Find the general solutions of the following equations and compare computer solutions.

${\mathit{D}}^{\mathbf{2}}{\mathbf{(}\mathbf{D}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}{\mathbf{(}\mathbf{D}\mathbf{+}\mathbf{2}\mathbf{)}}^{\mathbf{3}}\mathit{y}\mathbf{=}\mathbf{0}$

Verify that is a particular solution of .Verify that another particular solution ofis

Observe that we obtain the same general solutionwhichever particular solution we use [sinceis just as good an arbitrary constant as. Show in general that the difference between two particular solutions ofis always a solution of the homogeneous equation, and thus show that the general solution is the same for all choices of a particular solution.

Find the transform of

$\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\{\begin{array}{ll}sin(x-vt),& t>x/v\\ 0,& t<x/v\end{array}$

Where xand vare constants.

For the following problems, verify the given solution and then, by method (e) above, find a second solution of the given equation.

$x{y}^{\text{'}\text{'}}-2(x+1)y^{\text{'}}+(x+2)y=0$

Solve the following sets of equations by the Laplace transform method

$$\begin{gathered} \dot{\mathbf{y}}\mathbf{+}\mathit{z}\mathbf{=}\mathbf{2}\mathit{c}\mathit{o}\mathit{s}\mathit{t}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathbf{y}}_{\mathbf{0}}\mathbf{=}\mathbf{-}\mathbf{1} \\ \dot{\mathbf{z}}\mathbf{-}\mathit{y}\mathbf{=}\mathbf{1}\mathbf{}\mathbf{}\mathbf{}{\mathit{z}}_{\mathbf{0}}\mathbf{=}\mathbf{1} \end{gathered}$$

For the following problems, verify the given solution and then, by method (e) above, find a second solution of the given equation

$3x{y}^{\text{'}\text{'}}-2(3x-1)y^{\text{'}}+(3x-2)y=0$