Exact Differential Equations and Integrating Factors Cheat Sheet

Key Facts

• A first order equation in differential form is written as $M(x,y)\,dx + N(x,y)\,dy = 0$.
• The equation $M(x,y)\,dx + N(x,y)\,dy = 0$ is exact when $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$ on a region where the needed derivatives are continuous.
• For an exact equation, find a potential function $\psi(x,y)$ satisfying $\psi_x=M$ and $\psi_y=N$.
• The implicit general solution of an exact equation is $\psi(x,y)=C$, where $C$ is an arbitrary constant.
• A common way to build $\psi$ is $\psi(x,y)=\int M(x,y)\,dx+g(y)$, then use $\psi_y=N$ to find $g'(y)$.
• If $\frac{M_y-N_x}{N}=f(x)$ depends only on $x$, then an integrating factor is $\mu(x)=e^{\int f(x)\,dx}$.
• If $\frac{N_x-M_y}{M}=g(y)$ depends only on $y$, then an integrating factor is $\mu(y)=e^{\int g(y)\,dy}$.
• For a linear equation $\frac{dy}{dx}+P(x)y=Q(x)$, the integrating factor is $\mu(x)=e^{\int P(x)\,dx}$.

Vocabulary

Differential form

A first order differential equation written as $M(x,y)\,dx+N(x,y)\,dy=0$.

Exact equation

An equation $M\,dx+N\,dy=0$ is exact when it can be written as $d\psi=0$ for some potential function $\psi(x,y)$.

Potential function

A function $\psi(x,y)$ whose differential is $d\psi=\psi_x\,dx+\psi_y\,dy$.

Integrating factor

A nonzero function $\mu$ that multiplies a differential equation to make it exact or easier to integrate.

Implicit solution

A solution written as a relation such as $\psi(x,y)=C$ instead of solving explicitly for $y$.

Exactness condition

The test $M_y=N_x$ used to determine whether $M(x,y)\,dx+N(x,y)\,dy=0$ is exact.