Separable differential equations

What you’ll learn

• Separate variables and integrate to obtain a general solution.
• Use an initial condition to determine a particular solution.
• Interpret a solution and any equilibrium cases in context.

Understand the idea

An equation is separable when it can be written as $dy/dx=f(x)g(y)$. Move the y-dependent factor to the y side and integrate each side with respect to its own variable. One arbitrary constant is enough because the two constants from integration can be combined.

An initial condition determines that constant. If logarithms occur, handle absolute values or the stated sign of the quantity before exponentiating. A general solution describes a family; a particular solution satisfies the given starting condition.

Dividing by $g(y)$ can remove constant solutions where $g(y)=0$, so check these separately. Interpret a model's growth, decay or limiting value and the time range in which its assumptions make sense. Rates described in words must use the same variables and units as the resulting equation.

Key formulae

$$\frac{dy}{dx}=f(x)g(y)\Rightarrow\int\frac1{g(y)}\,dy=\int f(x)\,dx$$

$$\frac{dy}{dt}=ky\Rightarrow y=Ae^{kt}$$

$$\frac{dy}{dt}=k(L-y)\Rightarrow y=L+Ae^{-kt}$$

Worked example

Solve $dy/dx=2xy$ given $y=3$ when $x=0$.

1. For the non-zero solution, separate: $(1/y)\,dy=2x\,dx$.
2. Integrate to get $\ln|y|=x^2+C$, so $y=Ae^{x^2}$.
3. The condition gives $3=Ae^0$, so $A=3$. The zero solution does not satisfy this initial condition.

Answer: $y=3e^{x^2}$.