Chapter 8: Ordinary Differential Equations

Find the shape of a mirror which has the property that rays from a point 0 on the axis are reflected into a parallel beam. Hint: Take the point 0 at the origin. Show from the figure that $\mathit{t}\mathit{a}\mathit{n}\mathbf{}\mathit{\theta }\mathbf{=}\mathbf{2}\frac{\mathbf{y}}{\mathbf{x}}$. Use the formula for $\mathit{t}\mathit{a}\mathit{n}\mathit{\theta }\mathbf{2}$to express this in terms of $\mathit{t}\mathit{a}\mathit{n}\mathit{\theta }\mathbf{=}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}$and solve the resulting differential equation. (Hint: See Problem 16.)

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

${\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{\text{'}}\mathbf{-}\mathbf{8}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{16}\mathbf{y}\mathbf{=}\mathbf{32}\mathbf{t}\mathbf{,}\mathbf{\hspace{1em}}{\mathbf{y}}_{\mathbf{0}}\mathbf{=}\mathbf{1}\mathbf{,}{\mathbf{y}}_{\mathbf{0}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{2}$

Use L29 to verify L6, L13, L14, and L18

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 $\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{26}\mathbf{)}\mathbf{,}$andthe discussion after$\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{15}\mathbf{)}\mathbf{,}$].

${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{4}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{8}\mathit{y}\mathbf{=}\mathbf{30}{\mathit{e}}^{\mathbf{-}\mathbf{x}\mathbf{/}\mathbf{2}}\mathit{c}\mathit{o}\mathit{s}\mathbf{5}\mathit{x}\mathbf{/}\mathbf{2}$

Solve the following equations using method (d) above.

$x^2{y}^{\text{'}\text{'}}-3x{y}^{\text{'}}+4y=6x^2\ln x$

Suppose the rate at which bacteria in a culture grow is proportional to the number present at any time. Write and solve the differential equation for the number N of bacteria as a function of time t if there are ${\mathit{N}}_{\mathbf{0}}$bacteria when $\mathit{t}\mathbf{=}\mathbf{0}$. Again note that (except for a change of sign) this is the same differential equation and solution as in the preceding problems.

Solve the following equations using method (d) above.

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

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

$\ddot{\mathbf{y}}\mathbf{+}\mathbf{4}\dot{\mathbf{y}\mathbf{+}\mathbf{5}\mathbf{y}\mathbf{=}\mathbf{26}{\mathbf{e}}^{\mathbf{3}\mathbf{t}}\mathbf{,}}\mathbf{}\mathbf{}{\mathit{y}}_{\mathbf{0}}\mathbf{=}\mathbf{1}\mathbf{,}\dot{{\mathbf{y}}_{\mathbf{0}}\mathbf{=}\mathbf{5}}$

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

$\mathbf{5}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathbf{6}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{6}\mathit{x}$

Use L29 and L11 to obtain $\mathit{L}\left(t{e}^{-at}sinbt\right)$which is not in the table.