Differential Equations First- and Second-Order Cheat Sheet

Key Facts

• A separable first-order equation has the form $\frac{dy}{dx} = g(x)h(y)$ and is solved by rewriting it as $\frac{1}{h(y)}\,dy = g(x)\,dx$ before integrating.
• A linear first-order equation has the form $y' + P(x)y = Q(x)$ and uses the integrating factor $\mu(x) = e^{\int P(x)\,dx}$.
• After multiplying by the integrating factor, $y' + P(x)y = Q(x)$ becomes $\frac{d}{dx}\left[\mu(x)y\right] = \mu(x)Q(x)$.
• A homogeneous second-order linear equation with constant coefficients has the form $ay'' + by' + cy = 0$ and characteristic equation $ar^2 + br + c = 0$.
• If the characteristic roots are distinct real numbers $r_1$ and $r_2$, then the general solution is $y = C_1e^{r_1x} + C_2e^{r_2x}$.
• If the characteristic equation has a repeated real root $r$, then the general solution is $y = C_1e^{rx} + C_2xe^{rx}$.
• If the characteristic roots are complex $r = \alpha \pm \beta i$, then the general solution is $y = e^{\alpha x}\left(C_1\cos(\beta x) + C_2\sin(\beta x)\right)$.
• For a nonhomogeneous equation $ay'' + by' + cy = f(x)$, the general solution is $y = y_h + y_p$, where $y_h$ solves the homogeneous equation and $y_p$ is one particular solution.

Vocabulary

Differential equation

An equation involving an unknown function and one or more of its derivatives.

Order

The order of a differential equation is the highest derivative that appears, such as first order for $y'$ or second order for $y''$.

Initial condition

A value such as $y(0)=2$ or $y'(0)=-1$ that is used to determine constants in a general solution.

Integrating factor

A function $\mu(x) = e^{\int P(x)\,dx}$ that turns a first-order linear equation into an exact derivative.

Characteristic equation

The algebraic equation $ar^2 + br + c = 0$ obtained from a constant-coefficient homogeneous equation $ay'' + by' + cy = 0$.

Particular solution

A single solution $y_p$ that satisfies a nonhomogeneous differential equation such as $ay'' + by' + cy = f(x)$.