Problems

01

Decide whether the ODE is separable, first-order linear, or exact, and then solve it:

• (a) $y^{\prime }y=x+1$
• (b) $(2y-x^3)dx=xdy$
• (c) $(2x{y}^4+\sin y)dx+(4x^2{y}^3+x\cos y)dy=0$
• (d) $y-x+xy\cot (x)+x{y}^{\prime }=0$
• (e) $xy{y}^{\prime }=y-1$
• (f) $y\ln ydx-xdy=0$

02

Exact equation: 2D gradient emanating from charged particle

Consider an electric field $\mathbf{F}$ in ${\mathbb{R}}^2$ given by:

$$(F_1,F_2)=\left(\frac{x}{(x^2+y^2{)}^{3/2}},\frac{y}{(x^2+y^2{)}^{3/2}}\right).$$

Find a scalar function $f(x,y)$ whose gradient is $\mathbf{F}$. What are the level curves of $f$? Now write an exact differential equation whose integral curves are these level curves of $f$.