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.

${\mathbf{y}}^{\mathbf{\text{'}\text{'}}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{[}{\mathbf{x}}^3\mathbf{-}\mathbf{1}\mathbf{]}\mathbf{+}\mathbf{\left[}\mathbf{2}\mathbf{cosx}\mathbf{\right]}\mathbf{+}\mathbf{\left[}\mathbf{(}\mathbf{2}\mathbf{-}\mathbf{4}\mathbf{x}\mathbf{)}{\mathbf{e}}^x\mathbf{\right]}$

Find the equation of motion of a simple pendulum (see Chapter 7, Problem 2.13), that is, the differential equation for$\theta$ Vas a function of$t$. Show that, for small,$\theta$ this is approximately a simple harmonic motion equation, and find$\theta$if$\theta ={\theta }_0,d\theta /dt=0$when$t=0$.

Using Problems 29 and 31b, show that equation (6.24) is correct.

Several Terms on the Right-Hand Side: Principle of Superposition So far we have brushed over a question which may have occurred to you: What do we do if there are several terms on the right-hand side of the equation involving different exponentials?

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.

${\mathbf{y}}^{\mathbf{"}}\mathbf{-}\mathbf{5}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{=}\mathbf{2}{\mathbf{e}}^{\mathbf{x}}\mathbf{+}\mathbf{6}\mathbf{x}\mathbf{-}\mathbf{5}$

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

${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}{\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}\mathit{s}\mathit{i}\mathit{n}\mathbf{3}\mathit{t}\mathit{d}\mathit{t}\mathbf{=}\frac{\mathbf{3}}{\mathbf{13}}$. Hint: In (8.1), let p=2,f(t) =sin3t;use L3 with a=3.

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

${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}\mathit{t}{\mathit{e}}^{\mathbf{-}\mathbf{t}}\mathit{s}\mathit{i}\mathit{n}\mathbf{5}\mathbf{tdt}$

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.

$\mathbf{(}{\mathbf{D}}^{\mathbf{2}}\mathbf{-}\mathbf{1}\mathbf{)}\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\mathbf{}\mathit{x}$

The gravitational force on a particle of mass$m$inside the earth at a distance$r$from the centre$(r<$the radius of the earth$R$) Is$F=-mgr/R$(Chapter 6, Section 8, Problem 21). Show that a particle placed in an evacuated tube through the centre of the earth would execute simple harmonic motion. Find the period of this motion.

Find the orthogonal trajectories of each of the following families of curves. In each case, sketch or computer plot several of the given curves and several of their orthogonal trajectories. Be careful to eliminate the constant from $\mathit{y}\mathbf{\text{'}}$for the original curves; this constant takes different values for different curves of the original family, and you want an expression for $\mathit{y}\mathbf{\text{'}}$which is valid for all curves of the family crossed by the orthogonal trajectory you are trying to find. See equations $\left(2.10\right)$to $\left(2.12\right)$.

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

Find (in terms of$L$and$C$) the frequency of electrical oscillations in a series circuit (Figureif$\mathbf{R}=\mathbf{0}$or$V=0$, but$I\ne 0$. (When you tune a radio, you are adjusting$C$and/or$L$to make this frequency equal to that of the radio station.)

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

${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}\frac{{\mathbf{e}}^{\mathbf{-}\mathbf{3}\mathbf{t}}\mathbf{}\mathbf{sin}\mathbf{2}\mathbf{t}}{\mathbf{t}}\mathbf{dt}$