9 Ordinary Differential Equations I

1. For the ODE
   
   $$y' = x(1-y)$$
   
   
   sketch a dense direction field as a fully covering set of tiny line segments. Based on that, draw various solution curves. Discuss maxima and minima, symmetry, asymptotes, and inflection points of the solutions. Do not solve the equation algebraically. Use only the direction field to derive the solution properties.

2. Solve
   
   $$2 y \frac{{\rm d} y}{{\rm d} x} = e^x e^{-y^2} \qquad y(4)=-2$$
   
   
   Now solve the same ODE, but with initial condition that $y=0$ at $x=1$. Accurately draw these solutions.

3. Solve, using the class procedure (variation of parameter),
   
   $$\sin(2x) \frac{{\rm d} y}{{\rm d} x} = 2 \sin(x) - 2 y \sin^2(x)$$
   
   

4. Solve
   
   $$2 y'' - 4 y' + 20 y = 0$$
   
   
   As always, don't forget to clean up.

5. Solve
   
   $$2 y'' - 4 y' - 6 y = 4 \sin^2(x)$$
   
   
   Use variation of parameters.