First-Order Separable ODEs

≈ 25 min

A first-order ODE is separable if it can be written so that each side involves only one variable:

\frac{dy}{dx} = g(x)\,h(y) \quad\Longrightarrow\quad \frac{dy}{h(y)} = g(x)\,dx.

Integrate both sides and solve for $y$ (a constant of integration always appears).

Worked example. Solve $\dfrac{dy}{dx} = xy.$

\int \frac{dy}{y} = \int x\, dx \;\Rightarrow\; \ln|y| = \tfrac{x^2}{2} + C \;\Rightarrow\; y = A\,e^{x^2/2}.

The important special case $\dfrac{dy}{dx} = ky$ gives exponential growth or decay, $y = C e^{kx}.$

Which of these first-order ODEs is separable?

$\dfrac{dy}{dx} = xy$

$\dfrac{dy}{dx} = x + y$

$\dfrac{dy}{dx} = \sin(x + y)$

$\dfrac{dy}{dx} = x^2 + y^2$

Solve $\dfrac{dy}{dx} = \dfrac{x}{y}$ with $y(0) = 3$ , and find $y$ when $x = 4$ (take the positive root).

What is the general solution of $\dfrac{dy}{dx} = ky$ (with $k$ constant)?

$y = C e^{kx}$

$y = kx + C$

$y = C e^{kx^2}$

$y = \dfrac{C}{x}$

Solve $\dfrac{dy}{dx} = y^2$ with $y(0) = 1$ , and find $y$ when $x = 0.5$ .