Chapter 8: Ordinary Differential Equations

Using thefunction method, find the response (see Problem fig) of each of the following systems to a unit impulse.

$\mathit{y}\mathbf{"}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{10}\mathit{y}\mathbf{=}\mathit{\delta }\left(t-t_0\right)$

Using $\left(3.9\right)$, find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after $\mathbf{(}\mathbf{3}\mathbf{.}\mathbf{9}\mathbf{)}$, and Example 1.

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

Use the convolution integral to find the inverse transforms of:

$\frac{\mathbf{2}}{{\mathbf{p}}^{\mathbf{3}}\left(p+2\right)}$

For each of the following differential equations, separate variables and find a solution containing one arbitrary constant. Then find the value of the constant to give a particular solution satisfying the given boundary condition. Computer plot a slope field and some of the solution curves.

9 $\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{y}\mathbf{)}\mathbf{y}\mathbf{\text{'}}\mathbf{=}\mathbf{y}\mathbf{,}$ $\mathbf{y}\mathbf{=}\mathbf{1}$When$\mathbf{x}\mathbf{=}\mathbf{1}$