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Double and complex root

When the characteristic equation has a double root, the second solution needs an extra factor of $x$ to be independent of the first. When the roots are complex, $\pm i\beta$ , the solution becomes sine and cosine instead of exponentials.

(C_1 + C_2x)e^{rx}

double root

r

C_1\cos\beta x + C_2\sin\beta x

purely imaginary roots

\pm i\beta

Symbols

$r$	the double root
$\beta$	imaginary part (oscillation frequency)

Example

$y''+y=0$ has roots $r=\pm i$ : $y=C_1\cos x+C_2\sin x$ .

Double root $r=2$ : $y=(C_1+C_2x)e^{2x}$ .

A common mistake is writing

C_1e^{rx}+C_2e^{rx}

for a double root - that is only one solution, not two.

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Part of Linear Algebra and Differential Equations: Differential equations.