Chapter 8: Ordinary Differential Equations

Question: Identify each of the differential equations in Problems 1to 24 as to type (for example, separable, linear first order, linear second order, etc.), and then solve it.

Question: Identify each of the differential equations in Problems 1to 24 as to type (for example, separable, linear first order, linear second order, etc.), and then solve it.

Question: Identify each of the differential equations in Problems 1to 24 as to type (for example, separable, linear first order, linear second order, etc.), and then solve it.

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

$\mathbf{\hspace{0.33em}}\mathbf{\hspace{0.33em}}\mathbf{y}\mathbf{"}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{5}\mathbf{sinh}\mathbf{}\mathbf{2}\mathbf{t}\mathbf{,}\mathbf{\hspace{0.33em}}\mathbf{\hspace{0.33em}}\mathbf{\hspace{0.33em}}\mathbf{y}\mathbf{0}\mathbf{=}\mathbf{0}\mathbf{,}{\mathbf{\psi }}_{\mathbf{0}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{2}$

Find the inverse transforms of the functions$F\left(p\right)$.

$\frac{6-p}{p^2+4p+20}$

Find the general solution of the following differential equations (complementary function + particular solution). Find the particular solution by inspection. Also find a computer solution and reconcile differences if necessary, noticing especially whether the particular solution is in simplest form.

$\mathbf{(}{\mathbf{D}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathbf{D}\mathbf{+}\mathbf{1}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{2}\mathit{c}\mathit{o}\mathit{s}\mathit{x}$

Use the Laplace transform table to find $\mathrm{f}\left(\mathrm{t}\right)={\int }_0^{\mathrm{t}}{\mathrm{e}}^{-\mathrm{\tau }}\sin \left(\mathrm{t}-\mathrm{\tau }\right)\mathrm{d\tau }$. Hint: In L34, let$\mathit{g}\left(t\right)\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{t}}$and$\mathit{h}\left(t\right)\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{t}$, and find$\mathit{G}\left(p\right)\mathit{H}\left(p\right)$which is the Laplace transform of the integral you want. Break the result into partial fractions and look up the inverse transforms.

The exact equation of motion of a simple pendulum is$d^2\theta /d{t}^2=-{\omega }^{2}sin\theta$where${\omega }^2=g/l$. By method (c) above, integrate this equation once to find$d\theta /dt$if$d\theta /dt=0$when$\theta =90°$. Write a formula for$t\left(\theta \right)$as an integral. See Problem 5.34.

In Problems 13 to 15, find a solution (or solutions) of the differential equation not obtainable by specializing the constant in your solution of the original problem. Hint: See Example 3.

13. Problem 2

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

$\mathit{y}\mathit{y}\mathbf{\text{'}}\mathbf{-}\mathbf{2}{\mathit{y}}^{\mathbf{2}}\mathit{c}\mathit{o}\mathit{t}\mathit{x}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{x}$