Chapter 8: Ordinary Differential Equations

Use the convolution integral (see Example 2) to solve the following differential equations. $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{=}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}\mathbf{,}\mathbf{}{\mathbf{y}}_{\mathbf{0}}\mathbf{=}{\mathbf{y}}_{\mathbf{0}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{0}$.

Find the general solution of the following differential equations (complementary function + particular solution). Find the particular solution by inspection. Also find a computer solution and reconcile differences if necessary, noticing especially whether the particular solution is in simplest form.

${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{8}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{25}\mathit{y}\mathbf{=}\mathbf{120}\mathit{s}\mathit{i}\mathit{n}\mathbf{5}\mathit{x}$

In problems 13 to 15, find a solution(or solutions) of the differential equation not obtainable by specializing the constant in your solution of the original problem. Hint: See Example 3.

14. Problem 8.

Show that a combination of entries L 3 to L 10, L 13, L 14 and L 18 in the table, will give the inverse transform of any function of the form$\frac{Ap+B}{C{p}^2+Ep+F}$

, where A, B, C, E, andare constants.

Verify (7.19) and (7.20).; write the first equation of (7.19) as $x{D}_X=D_Z$, and find ${D^2}_Z$.

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.

$\mathbf{y}\mathbf{"}\mathbf{-}\mathbf{4}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{-}\mathbf{4}{\mathbf{te}}^{\mathbf{2}}\mathbf{t}\mathbf{,}\mathbf{\hspace{1em}}{\mathbf{y}}_{\mathbf{0}}\mathbf{=}\mathbf{0}\mathbf{,}{\mathbf{y}}_{\mathbf{0}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{1}$

Prove L32 for$n=1$. Hint: Differentiate equation (8.1) with respect to$p$.

Use the convolution integral (see Example 2) to solve the following differential equations

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathbf{y}\mathbf{=}{\mathbf{e}}^{\mathbf{3}\mathbf{t}}\mathbf{,}\mathbf{}{\mathbf{y}}_{\mathbf{0}}\mathbf{=}{\mathbf{y}}_{\mathbf{0}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{0}$

Find the general solution of the following differential equations (complementary function + particular solution). Find the particular solution by inspection or by or .Also find a computer solution and reconcile differences if necessary, noticing especially whether the particular solution is in simplest form [seeand the discussion after].

$\mathbf{5}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{12}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{20}\mathit{y}\mathbf{=}\mathbf{120}\mathit{s}\mathit{i}\mathit{n}\mathbf{2}\mathit{x}$

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.

${\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathbf{y}\mathbf{=}\mathbf{cos}\mathbf{3}\mathbf{t}\mathbf{,}\mathbf{\hspace{1em}}{\mathbf{y}}_{\mathbf{0}}\mathbf{=}\mathbf{0}\mathbf{,}{\mathbf{y}}_{\mathbf{0}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{6}$