91Ó°ÊÓ

$\mathbf{x}\mathbf{+}\mathbf{y}$(a) Prove that $\mathbf{Mdx}\mathbf{+}\mathbf{Ndy}\mathbf{=}\mathbf{0}$has an integrating factor that depends only on the sum x + y if and only if the expression $\frac{\left(\frac{∂\mathbf{N}}{∂\mathbf{x}}\mathbf{-}\frac{∂\mathbf{M}}{∂\mathbf{y}}\right)}{\left(\mathbf{M}\mathbf{-}\mathbf{N}\right)}$depends only on .

(b) Use part (a) to solve the equation $\left(\mathbf{3}\mathbf{+}\mathbf{y}\mathbf{+}\mathbf{xy}\right)\mathbf{dx}\mathbf{+}\left(\mathbf{3}\mathbf{+}\mathbf{x}\mathbf{+}\mathbf{xy}\right)\mathbf{dy}\mathbf{=}\mathbf{0}$.

Question: Use the substitution $\mathbf{v}\mathbf{=}\mathbf{x}\mathbf{-}\mathbf{y}\mathbf{+}\mathbf{2}$ to solve equation (8).

$\frac{\mathrm{dy}}{\mathrm{dx}}\mathbf{=}\mathbf{y}\mathbf{-}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{+}{\left(x-y+2\right)}^{-1}$

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.)

In Problems , identify the equation as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone.

$\left({\mathbf{x}}^{\mathbf{2}}\mathbf{sinx}\mathbf{+}\mathbf{4}\mathbf{y}\right)\mathbf{dx}\mathbf{+}\mathbf{xdy}\mathbf{=}\mathbf{0}$