Chapter 7: The Laplace Transform

Question: In problems31 and 32use the Laplace transform And the procedure outlined in exampleto solve the given boundary-value problem.

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{+}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{,}\mathit{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{0}$

Use the procedures developed in this chapter to find the general solution of each differential equation.

${\mathit{x}}^{\mathbf{2}}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{-}\mathbf{4}\mathit{x}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{6}\mathit{y}\mathbf{=}\mathbf{2}{\mathit{x}}^{\mathbf{4}}\mathbf{+}{\mathit{x}}^{\mathbf{2}}$

In Problems 29–32 express f in terms of unit step functions. Find

and.

31.

Use theorem 7.1.1 to find$L\left\{f\left(t\right)\right\},$if$f\left(t\right)=\cos 5t+\sin 2t?$

Find the Laplace transform of f * g using Theorem 7.4.2. Do not evaluate the convolution integral before transforming.

$\mathbf{L}\left\{{\int }_0^t\mathrm{sinT}.\cos \left(t-T\right)\mathrm{dT}\right\}$

In problemsanduse the Laplace transform And the procedure outlined in exampleto solve the given boundary-value problem.

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{8}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{20}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathit{\pi }\mathbf{\right)}\mathbf{=}\mathbf{0}$

use the procedures developed in this chapter to find the general solution of each differential equation.

${\mathit{x}}^{\mathbf{2}}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{-}\mathit{x}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathit{y}\mathbf{=}{\mathit{x}}^{\mathbf{3}}$

In Problems 29–32 express f in terms of unit step functions. Find$\mathit{L}\left\{f\left(t\right)\right\}$

and $\mathit{L}\left\{e^{t}f\left(t\right)\right\}$
.

Use theorem 7.1.1 to find$L\left\{f\left(t\right)\right\},$if$f\left(t\right)=\sinh kt?$

Find the Laplace transform of f * g using Theorem 7.4.2. Do not evaluate the convolution integral before transforming.

$\mathbf{L}\left\{t{\int }_0^t\mathrm{sinTdT}\right\}$