Differential Equations (Advanced)

Harder items at competition-adjacent depth.

Phase 2 · AP Calculus

Differential Equations: core idea

Initial value problems.

Solve $y'+2y=e^x$ , $y(0)=0$ . Find $y(\ln 2)$ .

Hint: Integrating factor $e^{2x}$ . Answer: 7/12.

1. Solve $y'=y/x$ for $x>0$ , $y(1)=2$ . Find $y(4)$ .
2. Solve $y'=x(1+y^2)$ , $y(0)=0$ . Find $y(1)$ .
3. Phase line $y'=y(y-2)(y+1)$ : unstable equilibrium among $-1,0,2$ ? Enter it.
4. Solve $y'=\sin x/y$ with $y(0)=1$ . Find $y(\pi)^2$ .
5. Euler twice: $y'=y$ , $y(0)=1$ , $h=0.5$ : approx $y(1)$ ?
6. Solve $y''+y=0$ , $y(0)=0$ , $y'(0)=1$ . Find $y(\pi/2)$ .
7. Solve $y'=y\ln y$ , $y(0)=e$ . Find $y(1)$ .
8. Logistic carrying capacity if $y'=0.2y(1-y/50)$ ?
9. IVP $y'+y=1$ , $y(0)=0$ . Find $\lim_{x\to\infty}y(x)$ .