The characteristic equation and the three cases

Property — Characteristic equation

For $ay'' + by' + cy = 0$ we look for the solution in the form $y = e^{\lambda x}$, from which the characteristic equation: $a\lambda^2 + b\lambda + c = 0.$ Three cases, according to the sign of the discriminant $\Delta$:

• $\Delta > 0$: two real roots $\lambda_1,\lambda_2$ $\implies y = C_1 e^{\lambda_1 x} + C_2 e^{\lambda_2 x}$;
• $\Delta = 0$: double root $\lambda$ $\implies y = (C_1+C_2 x)e^{\lambda x}$;
• $\Delta < 0$: complex roots $\alpha\pm i\beta$ $\implies y = e^{\alpha x}(C_1\cos\beta x + C_2\sin\beta x)$.

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Topics: Differential equations
Concepts: Second-order linear ODE · Characteristic equation
Skills: Reasoning by cases · Solving equations