91Ó°ÊÓ

a) Show that$\mathbf{∇}\mathbf{â‹\dots }\mathbf{(}{\mathit{e}}_{\mathbf{r}}\mathbf{/}{\mathit{r}}^{\mathbf{2}}\mathbf{)}\mathbf{=}\mathbf{0}$for$\mathit{r}\mathbf{>}\mathbf{0}$.

(b) Show that$\mathbf{∇}\mathbf{(}\mathbf{1}\mathbf{/}\mathit{r}\mathbf{)}\mathbf{=}\mathbf{-}{\mathit{e}}_{\mathbf{r}}\mathbf{/}{\mathit{r}}^{\mathbf{2}}$.

Sketch on the same axes graphs of$\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{t}\mathbf{,}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\left(t-\mathrm{Ï€}/2\right)$, and$\mathit{s}\mathit{i}\mathit{n}\left(t+\mathrm{Ï€}/2\right)$, and observe which way the graph shifts. Hint: You can, of course, have your calculator or computer plot these for you, but it's simpler and much more useful to do it in your head. Hint: What values of $\mathit{t}$make the sines equal to zero? For an even simpler example, sketch on the same axes$\mathit{y}\mathbf{=}\mathit{t}\mathbf{,}\mathit{y}\mathbf{=}\mathit{t}\mathbf{-}\mathbf{Ï€}\mathbf{/}\mathbf{2}\mathbf{,}\mathbf{y}\mathbf{=}\mathbf{t}\mathbf{+}\mathbf{Ï€}\mathbf{/}\mathbf{2}$.

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

${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{−}\mathbf{6}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{9}\mathit{y}\mathbf{=}\mathbf{12}\mathit{x}{\mathit{e}}^{\mathbf{3}\mathbf{x}}$

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

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

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.$(\mathbf{6}.24),$ Alsofind a computer solution and reconcile differences if necessary, noticing especially whether the particular solution is in simplest form [see$(\mathbf{6}.26),$and the discussionafter$\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{15}\mathbf{)}\mathbf{,}$].

$\mathbf{(}\mathit{D}\mathbf{−}\mathbf{3}\mathbf{)}\mathbf{(}\mathit{D}\mathbf{+}\mathbf{1}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{16}{\mathit{x}}^{\mathbf{2}}{\mathit{e}}^{\mathbf{−}\mathbf{x}}$

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

$\stackrel{\mathbf{¨}}{\mathbf{y}}\mathbf{+}\mathbf{4}\stackrel{\mathbf{˙}}{\mathbf{y}}\mathbf{+}\mathbf{5}\mathit{y}\mathbf{=}\mathbf{2}{\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{t}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathit{y}}_{\mathbf{0}}\mathbf{=}\mathbf{0}\mathbf{,}{\stackrel{\mathbf{˙}}{\mathbf{y}}}_{\mathbf{0}}\mathbf{=}\mathbf{3}$

Use L28 to find the Laplace transform of

$\mathit{f}\left(t\right)\mathbf{=}\{\begin{array}{ll}sin\left(t-\mathrm{Ï€}/2\right),& t>\mathrm{Ï€}/2.\\ 0,& t<\mathrm{Ï€}/2.\end{array}$

Use L28 to find the Laplace transform of

$\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\{\begin{array}{ll}sin(t-\mathrm{Ï€}/2,& t>\mathrm{Ï€}/2\\ 0,& t<\mathrm{Ï€}/2\end{array}$

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

${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{−}\mathbf{3}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{−}\mathbf{9}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{−}\mathbf{5}\mathit{y}\mathbf{=}\mathbf{0}$

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

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