Bring the chapter together: put a linear equation in standard form and solve it with an integrating factor, test an equation for exactness and recover its potential, and reduce an equation to one of these by a substitution. No worked example sits above the answers.

A locksmith carries a ring of keys. Most of the work is looking at the lock and knowing which key to try first.

Decide first which kind of equation it is. Linear: divide by the coefficient of $y'$ to reach $y'+p(x)y=q(x)$, multiply by $\mu=e^{\int p\,dx}$ and integrate. Exact: check $M_{y}=N_{x}$, integrate $M$ in $x$ and match the result with $N$. Bernoulli: set $v=y^{1-n}$ and solve the linear equation for $v$.

• The constant of integration goes in before dividing by $\mu$, or it multiplies the wrong thing.
• A linear equation has its solution on the whole interval where $p$ and $q$ are continuous.
• $M_{y}=N_{x}$ is necessary for exactness, and sufficient on a region without holes.
• The solutions of an exact equation are the level curves $F(x,y)=C$ of its potential.