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Ordinary Differential Equations

Source lecture(s): PHY621 Lec5

Intuition

ODE rules tell us how a quantity changes continuously. Solutions respond to boundary and initial conditions.

Formal Definition

An ODE of order \(n\) involves derivatives up to \(f^{(n)}(x)\). Linear form: \(a_n(x)y^{(n)}+\cdots+a_0(x)y=g(x)\).

Mathematical Formulation

Second-order linear ODE: \(y''+p(x)y'+q(x)y=r(x)\). Homogeneous when \(r=0\).

Derivation

For constant coefficients, substitute \(y=e^{rx}\) to get the characteristic polynomial \(a_n r^n+\cdots+a_0=0\).

Worked Example

\(y''-3y'+2y=0\) gives \(r^2-3r+2=0\), so \(r=1,2\) and \(y=C_1e^x+C_2e^{2x}\).

Common Mistakes

  • Forgetting linearly independent solutions for repeated roots.
  • Adding \(\ln|x|\) for Cauchy-Euler without verifying \(x>0\).

Quiz

Q1. What is the general solution of \(y''+k^2y=0\)?

Answer

\(y=A\cos(kx)+B\sin(kx)\).