Differential Equations

dy/dx = ky gives y = Ce^kx; separate and integrate.

The idea

A differential equation relates an unknown function to its own derivative, and to solve it is to find every function satisfying the relation. Where an equation like $x^{2} = 9$ has numbers for solutions, a differential equation has functions — normally infinitely many, until an extra condition pins one down.

The first equation to know is

$\frac{dy}{dx} = ky,$

which says the quantity grows at a rate proportional to its current size: a population whose births scale with its size, a balance earning interest, or, with $k$ negative, a decaying sample. Its solutions are $y = Ce^{kx}$, since differentiating $Ce^{kx}$ brings down a factor of $k$ and returns the same function, exactly as the equation demands. Setting $x = 0$ shows $C$ is the starting amount, so a single measurement selects one solution from the family. Drawing at each point a short segment with the slope the equation prescribes there — a slope field — displays the whole family at once: every solution curve runs tangent to the segments it passes.

An equation is separable when it can be rearranged so that everything involving $y$ sits beside $dy$ and everything involving $x$ sits beside $dx$. Integrating each side separately then leaves a relation between $x$ and $y$ with no derivative in it.

Ways to work on it

• Walkthrough. Exponential models and separable equations.
• Practice. Find C for an exponential-growth initial value.
• Hardest. Solve a separable differential equation end to end.

Not sure where to start? Take the ten-question placement test.