Chapter 8: Ordinary Differential Equations

Question: In Problems 15to 18, use the given solutions of the homogeneous equation to find a particular solution of the given equation. You can do this either by the Green function formulas in the text or by the method of variation of parameters in Problem 14b.

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

Question: Use equation (12.6) to solve Problem 10.18.

Question: Obtain (12.6) by using the convolution integral to solve (12.1).

Question: For Problem 10.17, show (as in Problem 1) that the Green function is

$\mathit{G}\left(t,t^{\text{'}}\right)\mathbf{=}\{\begin{array}{ll}0,& 0<t<t^{\text{'}}\\ (1/a)sinha\left(t-t^{\text{'}}\right),& 0<t^{\text{'}}<t\end{array}$

Thus write the solution of Problem 10.17as an integral [similar to (12.6)] and evaluate it.

Question: Use the Green function of Problem 6 to solve ${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{-}{\mathit{a}}^{\mathbf{2}}\mathit{y}\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{t}}\mathbf{,}{\mathit{y}}_{\mathbf{0}}\mathbf{=}{\mathit{y}}_{\mathbf{0}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{0}\mathbf{.}$

Question: Solve the differential equation${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{2}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathit{y}\mathbf{=}\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{,}\mathit{y}\mathbf{0}\mathbf{=}{\mathit{y}}_{\mathbf{0}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{0}$, where$\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\{\begin{array}{ll}1,& 0<t<a,\\ 0,& t>a.\end{array}$

As in Problems 6 and 7, find the Green function for the problem and use it in equation (12.4). Consider the cases$\mathit{t}\mathbf{<}\mathit{a}$ and$\mathit{t}\mathbf{>}\mathit{a}$separately.

Question: Following the proof of (12.4), show that (12.9) gives a solution of (12.7).

In Problem 11, find $v\left(x\right)$if$v=0,x=1,$at$t=0$. Then write an integral for$t\left(x\right)$.

Use the methods of this section to solve the following differential equations. Compare computer solutions and reconcile differences.

$\mathit{y}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{y}}{\mathbf{x}}\mathbf{-}\mathit{t}\mathit{a}\mathit{n}\frac{\mathbf{y}}{\mathbf{x}}$

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.

$\mathbf{12}\mathbf{.}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{xy}\mathbf{)}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{0}$ y = 1When x = 1.