<aside> 💡 The characteristic equation of an ODE in the form $ay’’+by’+cy=0$ is $ar^2+br+c = 0$

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• We can solve the characteristic equation using the quadratic formula, or factor to find the roots
• Determine the form of the general solution based on whether the characteristic equation has distinct real roots, single repeated root, or complex conjugate roots.

Solutions to Characteristic Equations

Distinct Real Roots

• If $r_1$ and $r_2$ are real and distinct, we can write the general solution in the form: $y(x) = c_1e^{r_1x}+c_2e^{r_2x}$

Single Repeated Root

• If the roots are real and equal, the general solution is $y(x) = c_1e^{rx}+c_2xe^{rx}$

Complex Conjugate Roots

• If both roots are complex, the general solution is $y(x) = e^{\alpha x}(c_1 \cos \beta x + c_2 \sin \beta x)$
  • Characteristic equation has the complex conjugate roots $\alpha +- \beta i$