Second-Order Equations

By SkytrailGroup Mathematics Editorial Team April 12, 2026

Homogeneous

Characteristic equation: \(ar^2+br+c=0\).

Property	Statement
Two real roots	\(y=C_1e^{{r_1x}}+C_2e^{{r_2x}}\)
Repeated root	\(y=(C_1+C_2x)e^{{rx}}\)
Complex roots	\(y=e^{{\alpha x}}(C_1\cos\beta x+C_2\sin\beta x)\)

Examples

Example 1. y''−3y'+2y=0.

Solution. r²−3r+2=0 → r=1,2. y=C₁eˣ+C₂e²ˣ.

In Depth

Second-order ODEs \(F(x,y,y',y'')=0\) arise naturally in mechanics (Newton's second law), circuit theory, and wave propagation. The general solution of a second-order linear ODE contains two arbitrary constants, determined by two initial conditions (initial value problem) or two boundary conditions (boundary value problem).

The Wronskian \(W(y_1,y_2) = y_1y_2'-y_2y_1'\) tests linear independence of two solutions: if \(W\neq0\) on an interval, \(y_1\) and \(y_2\) form a fundamental set of solutions. Abel's theorem gives \(W(x) = W(x_0)e^{-\int_{x_0}^x P(t)\,dt}\) for \(y''+P(x)y'+Q(x)y=0\).

Boundary value problems (BVPs) specify conditions at two different points. Unlike IVPs, BVPs may have no solution, a unique solution, or infinitely many solutions. Sturm–Liouville theory studies a class of BVPs whose solutions form orthogonal bases — the foundation of Fourier series and spectral methods.

Euler–Cauchy equations \(x^2y''+axy'+by=0\) have solutions of the form \(y=x^r\). The substitution \(x=e^t\) converts them to constant-coefficient equations. They arise in problems with spherical or cylindrical symmetry.

Key Properties & Applications

The characteristic equation \(ar^2+br+c=0\) determines the behavior of solutions to \(ay''+by'+cy=0\). The discriminant \(\Delta=b^2-4ac\) classifies: \(\Delta>0\) (overdamped, two real roots), \(\Delta=0\) (critically damped, repeated root), \(\Delta<0\) (underdamped, complex roots with oscillation).

Resonance occurs in the forced equation \(y''+\omega_0^2y=F\cos(\omega t)\) when the driving frequency \(\omega\) equals the natural frequency \(\omega_0\). The particular solution grows like \(t\sin(\omega_0 t)\) — unbounded oscillation. Damping prevents true resonance but produces a maximum response near \(\omega_0\).

The Laplace transform converts a linear ODE with constant coefficients into an algebraic equation: \(\mathcal{L}\{y'\}=sY-y(0)\), \(\mathcal{L}\{y''\}=s^2Y-sy(0)-y'(0)\). This handles initial conditions automatically and is especially useful for piecewise or impulsive forcing functions.