Chapter 8: Ordinary Differential Equations

Solve the following equations using method (d) above.

$x^2{y}^{\text{'}\text{'}}+x{y}^{\text{'}}+y=2x$

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

$\ddot{\mathbf{y}}\mathbf{+}\mathbf{2}\dot{\mathbf{y}}\mathbf{+}\mathbf{5}\mathit{y}\mathbf{=}\mathbf{10}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{t}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}{\mathit{y}}_{\mathbf{0}}\mathbf{=}\mathbf{2}\mathbf{,}{\dot{\mathbf{y}}}_{\mathbf{0}}\mathbf{=}\mathbf{1}$

Solve the equation for the rate of growth of bacteria if the rate of increase is proportional to the number present but the population is being reduced at a constant rate by the removal of bacteria for experimental purposes

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

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

Obtain $\mathit{L}\left(t{e}^{-at}cosbt\right)$

Use the results which you have obtained in Problems 21 and 22 to find the inverse transform of$\left(p^2+2p-1\right)\mathbf{}\mathit{I}\mathbf{}{\left(p^2+4p+5\right)}^{\mathbf{2}}$.

Solve the two differential equations in Problem $5.11$of Chapter 13

$r\frac{d}{dr}\left(r\left(\frac{dR}{dr}\right)\right)=n^{2}R$

${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathit{y}\mathbf{=}\mathbf{2}\mathit{x}{\mathit{e}}^{\mathbf{x}}$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 $\mathbf{,}\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{26}\mathbf{)}\mathbf{,}$andthe discussion after$\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{15}\mathbf{)}\mathbf{,}$].${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathit{y}\mathbf{=}\mathbf{2}\mathit{x}{\mathit{e}}^{\mathbf{x}}$

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Heat is escaping at a constant rate [$\frac{\mathbf{d}\mathbf{Q}}{\mathbf{d}\mathbf{t}}$in $\left(1.1\right)$is constant] through the walls of a long cylindrical pipe. Find the temperature T at a distance r from the axis of the cylinder if the inside wall has radius $\mathit{r}\mathbf{=}\mathbf{1}$and temperature $\mathit{T}\mathbf{=}\mathbf{100}$and the outside wall has $\mathit{r}\mathbf{=}\mathbf{2}$and $\mathit{T}\mathbf{=}\mathbf{0}$

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$\mathbf{(}\mathbf{6}\mathbf{.15}\mathbf{)}$].

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