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Second order, real roots

For a homogeneous second-order equation $y''+ay'+by=0$ with constant coefficients we set up the characteristic equation. If it has two distinct real roots, the general solution is a sum of two exponentials.

r^2 + ar + b = 0

characteristic equation for

y''+ay'+by=0

y = C_1e^{r_1x} + C_2e^{r_2x}

general solution, two distinct real roots

Symbols

r_1,\ r_2

roots of the characteristic equation

Example

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

Set up the characteristic equation by replacing

y''\to r^2

y'\to r

y\to1

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Part of Linear Algebra and Differential Equations: Differential equations.