91Ó°ÊÓ

Question: (a) Show that the equation $\frac{\mathrm{dy}}{\mathrm{dx}}\mathbf{=}\mathbf{f}\left(x,y\right)$is homogeneous if and only if $\mathbf{f}\left(\mathrm{tx},\mathrm{ty}\right)\mathbf{=}\mathbf{f}\left(x,y\right)$.

(b) A function$\mathbf{H}\left(x,y\right)$is called homogeneous of order n if .$\mathbf{H}\left(\mathrm{tx},\mathrm{ty}\right)\mathbf{=}{\mathbf{t}}^n\mathbf{H}\left(x,y\right)$

Show that the equation$\mathbf{M}\left(x,y\right)\mathbf{dx}\mathbf{+}\mathbf{N}\left(x,y\right)\mathbf{dy}\mathbf{=}\mathbf{0}$is homogeneous if$\mathbf{M}\left(x,y\right)$and$\mathit{N}\left(x,y\right)$ are both homogeneous of the same order.

Question: Show that equation (13) reduces to an equation of the form $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}\mathit{G}\left(ax+by\right)\mathbf{,}$

when [Hint: If ${\mathit{a}}_{\mathbf{1}}{\mathit{b}}_{\mathbf{2}}\mathbf{=}{\mathit{a}}_{\mathbf{2}}{\mathit{b}}_{\mathbf{1}}$, then ${\mathit{a}}_{\mathbf{2}}\mathbf{/}{\mathit{a}}_{\mathbf{1}}\mathbf{=}{\mathit{b}}_{\mathbf{2}}\mathbf{/}{\mathit{b}}_{\mathbf{1}}\mathbf{=}\mathit{k}\mathbf{,}$so that${\mathit{a}}_{\mathbf{2}}\mathbf{=}\mathit{k}{\mathit{a}}_{\mathbf{1}}$ and ${\mathit{b}}_{\mathbf{2}}\mathbf{=}\mathit{k}{\mathit{b}}_{\mathbf{1}}$.]

Question: Coupled Equations. In analyzing coupled equations of the form

$\begin{array}{l}\frac{\mathrm{dy}}{\mathrm{dt}}\mathbf{=}\mathbf{ax}\mathbf{+}\mathbf{by}\\ \frac{\mathrm{dx}}{\mathrm{dt}}\mathbf{=}\mathbf{Î\pm ³\ae }\mathbf{+}\mathbf{β²â}\end{array}$

where a, b,$\mathbf{Î\pm }\text{ }\mathbf{and}\text{ â¶Ä‰}\mathbf{y}$ are constants, we may wish to determine the relationship between x and y rather than the individual solutions x(t), y(t). For this purpose, divide the first equation by the second to obtain

$\frac{\mathrm{dy}}{\mathrm{dx}}\mathbf{=}\frac{\mathrm{ax}+\mathrm{by}}{\mathrm{Î\pm ³\ae }+\mathrm{β²â}}$

This new equation is homogeneous, so we can solve it via the substitution $\mathbf{v}\mathbf{=}\frac{x}{y}$. We refer to the solutions of (17) as integral curves. Determine the integral curves for the system

$\begin{array}{l}\frac{\mathrm{dy}}{\mathrm{dt}}\mathbf{=}\mathbf{-}\mathbf{4}\mathbf{x}\mathbf{-}\mathbf{y}\\ \frac{\mathrm{dx}}{\mathrm{dt}}\mathbf{=}\mathbf{2}\mathbf{x}\mathbf{-}\mathbf{y}\end{array}$

Question: Magnetic Field Lines. As described in Problem 20 of Exercises 1.3, the magnetic field lines of a dipole satisfy.

$\frac{\mathrm{dy}}{\mathrm{dx}}\mathbf{=}\frac{3\mathrm{xy}}{\mathbf{2}{\mathbf{x}}^2\mathbf{-}{\mathbf{y}}^2}$

Solve this equation and sketch several of these lines.

Question: Riccati Equation. An equation of the form (18) $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}\mathit{P}\left(x\right){\mathit{y}}^{\mathbf{2}}\mathbf{+}\mathit{Q}\left(x\right)\mathit{y}\mathbf{+}\mathit{R}\left(x\right)$ is called a generalized Riccati equation.

1. If one solution—say, u(x)—of (18) is known, show that the substitution$\mathbf{y}\mathbf{=}\mathbf{u}\mathbf{+}\frac{1}{v}$reduces (18) to a linear equation in v.
2. Given that$\mathbf{u}\left(x\right)\mathbf{=}\mathbf{x}$is a solution to$\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}{\mathit{x}}^{\mathbf{3}}{\left(y-x\right)}^{\mathbf{2}}\mathbf{+}\frac{\mathbf{y}}{\mathbf{x}}\mathbf{,}$

use the result of part (a) to find all the other solutions to this equation. (The particular solution$\mathbf{u}\left(x\right)\mathbf{=}\mathbf{x}$can be found by inspection or by using a Taylor series method; see Section 8.1.)

Question: Derive the following general formula for the solution to

the Bernoulli equation (9):

$\begin{array}{c}\mathbf{y}\mathbf{=}{\left[\frac{\left(1-n\right){\displaystyle ∫e^{\left(1-n\right)∫P\left(x\right)dx}Q\left(x\right)}dx+C_1}{e^{\left(1-n\right){\displaystyle ∫P\left(x\right)dx}}}\right]}^{\raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{${}^{\mathbf{1}\mathbf{-}\mathbf{n}}$}\right.}\mathbf{,}\mathbf{f}\mathbf{o}\mathbf{r}\mathbf{n}\mathbf{≠}\mathbf{1}\\ \mathbf{y}\mathbf{=}{\mathbf{C}}_{\mathbf{2}}{\mathbf{e}}^{∫\left(Q\left(x\right)-P\left(x\right)\right)dx}\mathbf{,}\mathbf{f}\mathbf{o}\mathbf{r}\mathbf{n}\mathbf{=}\mathbf{1}\end{array}$

In problems, 1-8 identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form$y^{\text{'}}=G\left(ax+by\right)$.$\left(t+x+2\right)dx+\left(3t-x-6\right)dt=0$

In problems identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form .

In problems 1-8 identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form ${\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{G}\left(\mathrm{ax}+\mathrm{by}\right)$.$\left(t+x+2\right)\mathbf{dx}\mathbf{+}\left(3t-x-6\right)\mathbf{dt}\mathbf{=}\mathbf{0}$

In problems, 1-8 identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form ${\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{G}\left(\mathrm{ax}+\mathrm{by}\right)$.

$\mathbf{θ»å²â}\mathbf{-}\mathbf{²â»åθ}\mathbf{=}\sqrt{\mathbf{θ²â»åθ}}$