Problem 29
Find all functions \(M\) such that the equation is exact. (a) \(M(x, y) d x+\left(x^{2}-y^{2}\right) d y=0\) (b) \(M(x, y) d x+2 x y \sin x \cos y d y=0\) (c) \(M(x, y) d x+\left(e^{x}-e^{y} \sin x\right) d y=0\)
Problem 29
In Exercises \(25-29\) solve the initial value problem and sketch the graph of the solution. $$ \text { C/G } y^{\prime}+\frac{1}{x} y=\frac{2}{x^{2}}+1, \quad y(-1)=0 $$
Problem 30
In Exercises \(30-37\) solve the initial value problem. $$ (x-1) y^{\prime}+3 y=\frac{1}{(x-1)^{3}}+\frac{\sin x}{(x-1)^{2}}, \quad y(0)=1 $$
Problem 30
Solve the given homogeneous equation implicitly. $$ y^{\prime}=\frac{y^{3}+2 x y^{2}+x^{2} y+x^{3}}{x(y+x)^{2}} $$
Problem 30
According to Theorem \(2.1 .2,\) the general solution of the linear nonhomogeneous equation $$ y^{\prime}+p(x) y=f(x) $$ is $$ y=y_{1}(x)\left(c+\int f(x) / y_{1}(x) d x\right). $$ where \(y_{1}\) is any nontrivial solution of the complementary equation \(y^{\prime}+p(x) y=0 .\) In this exercise we obtain this conclusion in a different way. You may find it instructive to apply the method suggested here to solve some of the exercises in Section 2.1 . (a) Rewrite (A) as $$ [p(x) y-f(x)] d x+d y=0, $$ and show that \(\mu=\pm e^{\int p(x) d x}\) is an integrating factor for (C). (b) Multiply (A) through by \(\mu=\pm e^{\int p(x) d x}\) and verify that the resulting equation can be rewritten as $$ (\mu(x) y)^{\prime}=\mu(x) f(x) $$ Then integrate both sides of this equation and solve for \(y\) to show that the general solution of (A) is $$ y=\frac{1}{\mu(x)}\left(c+\int f(x) \mu(x) d x\right). $$ Why is this form of the general solution equivalent to (B)?
Problem 31
Solve the given homogeneous equation implicitly. $$ y^{\prime}=\frac{x+2 y}{2 x+y} $$
Problem 31
In Exercises \(30-37\) solve the initial value problem. $$ x y^{\prime}+2 y=8 x^{2}, \quad y(1)=3 $$
Problem 32
Prove: If the equations \(M_{1} d x+N_{1} d y=0\) and \(M_{2} d x+N_{2} d y=0\) are exact on an open rectangle \(R,\) so is the equation $$ \left(M_{1}+M_{2}\right) d x+\left(N_{1}+N_{2}\right) d y=0 $$
Problem 32
Solve the given homogeneous equation implicitly. $$ y^{\prime}=\frac{y}{y-2 x} $$
Problem 32
In Exercises \(30-37\) solve the initial value problem. $$ x y^{\prime}-2 y=-x^{2}, \quad y(1)=1 $$
