Second Order Odes

When to Use

Use this skill when working on second-order-odes problems in odes pdes.

Decision Tree

1. Classify the ODE

   • Constant coefficients: ay'' + by' + cy = f(x)?
   • Variable coefficients: y'' + P(x)y' + Q(x)y = R(x)?
   • Cauchy-Euler: x^2 y'' + bxy' + cy = 0?

2. Homogeneous with Constant Coefficients

   • Characteristic equation: ar^2 + br + c = 0
   • Distinct real roots: y = c1e^{r1x} + c2e^{r2x}
   • Repeated root: y = (c1 + c2x)e^{rx}
   • Complex roots a +/- bi: y = e^{ax}(c1cos(bx) + c2sin(bx))