91Ó°ÊÓ

In Exercises \(1-17\) determine which equations are exact and solve them. $$ \left(-2 y^{2} \sin x+3 y^{3}-2 x\right) d x+\left(4 y \cos x+9 x y^{2}\right) d y=0 $$

In Exercises \(6-11\) solve the initial value problem. $$ x y^{\prime}+\left(1+\frac{1}{\ln x}\right) y=0, \quad y(e)=1 $$

In Exercises \(1-13\) find all \(\left(x_{0}, y_{0}\right)\) for which Theorem 2.3 .1 implies that the initial value problem \(y^{\prime}=\) \(f(x, y), y\left(x_{0}\right)=y_{0}\) has (a) a solution (b) a unique solution on some open interval that contains \(x_{0}\). $$ y^{\prime}=\frac{2 x+3 y}{x-4 y} $$

In Exercises \(6-11\) solve the initial value problem. $$ x y^{\prime}+(1+x \cot x) y=0, \quad y\left(\frac{\pi}{2}\right)=2 $$

Find all solutions. Also, plot a direction field and some integral curves on the indicated rectangular region. $$ y^{\prime}\left(1+x^{2}\right)+x y=0 ;\\{-2 \leq x \leq 2,-1 \leq y \leq 1\\} $$

Find an integrating factor; that is a function of only one variable, and solve the given equation. $$ \left(27 x y^{2}+8 y^{3}\right) d x+\left(18 x^{2} y+12 x y^{2}\right) d y=0 $$

Solve the initial value problem. $$ y^{\prime}-x y=x y^{3 / 2}, \quad y(1)=4 $$

In Exercises \(1-17\) determine which equations are exact and solve them. $$ (2 x+y) d x+(2 y+2 x) d y=0 $$

In Exercises \(1-17\) determine which equations are exact and solve them. $$ \left(3 x^{2}+2 x y+4 y^{2}\right) d x+\left(x^{2}+8 x y+18 y\right) d y=0 $$

Solve the initial value problem. $$ x y^{\prime}+y=x^{4} y^{4}, \quad y(1)=1 / 2 $$