Second-Order Linear ODE Reference Cheat Sheet

Key Facts

• A second-order linear ODE has the standard form $a(x)y'' + b(x)y' + c(x)y = g(x)$, and it is homogeneous when $g(x) = 0$.
• For $ay'' + by' + cy = 0$, substitute $y = e^{rx}$ to get the characteristic equation $ar^2 + br + c = 0$.
• If the characteristic equation has distinct real roots $r_1$ and $r_2$, then $y_c = C_1e^{r_1x} + C_2e^{r_2x}$.
• If the characteristic equation has a repeated real root $r$, then $y_c = C_1e^{rx} + C_2xe^{rx}$.
• If the characteristic roots are complex $r = \alpha \pm \beta i$, then $y_c = e^{\alpha x}(C_1\cos(\beta x) + C_2\sin(\beta x))$.
• For a nonhomogeneous equation, the general solution is $y = y_c + y_p$, where $y_p$ is any one particular solution.
• The method of undetermined coefficients works best when $g(x)$ is built from polynomials, exponentials, sines, or cosines.
• Initial conditions such as $y(0) = y_0$ and $y'(0) = v_0$ are used after finding $y$ to solve for $C_1$ and $C_2$.

Vocabulary

Second-order ODE

A differential equation involving an unknown function and its derivatives up to the second derivative $y''$.

Linear differential equation

An equation where $y$, $y'$, and $y''$ appear only to the first power and are not multiplied together.

Homogeneous equation

A linear differential equation with zero forcing term, such as $ay'' + by' + cy = 0$.

Characteristic equation

The algebraic equation $ar^2 + br + c = 0$ formed from a constant-coefficient homogeneous ODE.

Complementary solution

The general solution $y_c$ of the associated homogeneous equation.

Particular solution

One specific solution $y_p$ of a nonhomogeneous equation $ay'' + by' + cy = g(x)$.