Second-Order Differential Equations

Level 1 - Calculus topic page in Differential Equations.

Principle

A second-order differential equation involves the second derivative of an unknown function. Such equations often model acceleration, oscillation, and curvature.

Notation

\(y''\)

second derivative of y

\(r\)

trial exponent parameter

\(C_1,C_2\)

constants in the general solution

The Core Method

For constant-coefficient homogeneous equations, try \(y=e^{rx}\). This gives a polynomial equation in \(r\).

Characteristic equation

\[ay''+by'+cy=0\Longrightarrow ar^2+br+c=0\]

The roots determine the form of the solution.

Worked Cases

Question

Solve \(y''-3y'+2y=0\).

Answer

The characteristic equation is \(r^2-3r+2=0\). Factor: \((r-1)(r-2)=0\), so \(r=1\) or \(r=2\). Therefore the general solution is \(y=C_1e^x+C_2e^{2x}\).

Examples

Question

What solution form follows from characteristic root \(r=4\) repeated twice?

Answer

A repeated root gives two independent solutions, \(e^{4x}\) and \(xe^{4x}\). The general solution is \(y=(C_1+C_2x)e^{4x}\).

First-Order Differential Equations

Series Solutions