Chapter 8: Ordinary Differential Equations

In Problem 33 to 38, solve the given differential equations by using the principle of superposition [see the solution of equation (6.29)]. For example, in Problem 33, solve three differential equations with right-hand sides equal to the three different brackets. Note that terms with the same exponential factor are kept together; thus, a polynomial of any degree is kept together in one bracket.

$\left(D^2+1\right)\mathbf{y}\mathbf{=}\mathbf{2}\mathbf{sin}\mathbf{}\mathbf{x}\mathbf{+}\mathbf{4}\mathbf{x}\mathbf{}\mathbf{cos}\mathbf{}\mathbf{x}$

A block of wood is floating in water; it is depressed slightly and then released to oscillate up and down. Assume that the top and bottom of the block are parallel planes which remain horizontal during the oscillations and that the sides of the block are vertical. Show that the period of the motion (neglecting friction) is$2\pi \sqrt{h/g}$ , where is the vertical height of the part of the block under water when it is floating at rest. Hint: Recall that the buoyant force is equal to the weight of displaced water.

In Problem 33 to 38, solve the given differential equations by using the principle of superposition [see the solution of equation (6.29)]. For example, in Problem 33, solve three differential equations with right-hand sides equal to the three different brackets. Note that terms with the same exponential factor are kept together; thus, a polynomial of any degree is kept together in one bracket.

${\left(D-1\right)}^{\mathbf{2}}\mathit{y}\mathbf{=}\mathbf{4}{\mathit{e}}^{\mathbf{x}}\mathbf{+}\left(1-x\right)\left(e^{2x}-1\right)$

In Problem 33 to 38, solve the given differential equations by using the principle of superposition [see the solution of equation (6.29)]. For example, in Problem 33, solve three differential equations with right-hand sides equal to the three different brackets. Note that terms with the same exponential factor are kept together; thus, a polynomial of any degree is kept together in one bracket.

${\mathit{y}}^{\mathbf{"}}\mathbf{-}\mathbf{2}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{9}\mathit{x}{\mathit{e}}^{\mathbf{-}\mathbf{x}}\mathbf{-}\mathbf{6}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}{\mathit{e}}^{\mathbf{2}\mathbf{x}}$

Evaluate each of the following definite integrals by using the Laplace transform table.

${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}{\mathbf{e}}^{\mathbf{-}\mathbf{t}}\mathbf{}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{cos}\mathbf{}\mathbf{2}\mathbf{t}\mathbf{)}\mathbf{dt}$

Evaluate each of the following definite integrals by using the Laplace transform table.

Find the solutions of (1.2)$\left(putI=dqIdt\right)$and (1.3), if $\mathit{V}\mathbf{=}{\mathit{V}}_{\mathbf{0}}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{\omega }\mathbf{\text{'}}\mathit{t}$( const.).

Using either$L2$or$L3$and$L4$verify$L9$and$L10$.

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.

3.$\mathbf{y}\mathbf{\text{'}}\mathbf{sinx}\mathbf{=}\mathbf{ylny}\mathbf{,}\mathbf{y}\mathbf{=}\mathbf{e}$when$\mathbf{x}\mathbf{=}\frac{\mathbf{\pi }}{\mathbf{3}}$

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.

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