How Do You Solve a Differential Equation by Separating Variables?

You solve a separable differential equation by rearranging it so that all terms involving the dependent variable are with its differential and all terms involving the independent variable are with its differential. You then integrate both sides and, if given, use an initial condition to determine the constant.

The Separation Method

For an equation of the form

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

divide by $g(y)$ and multiply by (dx):

$$\frac{1}{g(y)}\,dy=f(x)\,dx.$$

The variables are now separated. Integrate both sides:

$$\int \frac{1}{g(y)}\,dy=\int f(x)\,dx.$$

Include a single constant of integration $C$, then rearrange to make $y$ the subject if an explicit solution is required.

This method works because integration reverses differentiation on each separated side. The result before using an initial condition is the general solution, representing a family of curves. An initial condition selects one curve, producing a particular solution. If the equation cannot be written with one variable on each side, separation of variables is not appropriate.

Worked Example

Solve

$$\frac{dy}{dx}=3xy, \qquad y(0)=2.$$

Separate the variables:

$$\frac{1}{y}\,dy=3x\,dx.$$

Integrate both sides:

$$\ln|y|=\frac{3}{2}x^2+C.$$

Exponentiating gives

$$|y|=e^C e^{\frac{3}{2}x^2}.$$

Absorb the sign and $e^C$ into a new arbitrary constant $A$:

$$y=Ae^{\frac{3}{2}x^2}.$$

Use the initial condition $y(0)=2$:

$$2=Ae^0,$$

so $A=2$. Therefore,

$$\boxed{y=2e^{\frac{3}{2}x^2}}.$$

Verification is valuable: differentiating $y=2e^{\frac{3}{2}x^2}$ gives , and substituting gives . This confirms both the differential equation and the initial condition. Keep full algebraic accuracy throughout; unlike a numerical approximation, this answer is exact and does not require GDC rounding unless the question later asks for numerical evaluation.

A common misconception is that the constant must remain inside the logarithm. It may be rewritten after exponentiation because $e^C$ is simply another arbitrary constant. Also check whether dividing by a function of $y$ removes a constant solution, such as $y=0$.

IB Exam Technique

This is AHL 5.14, so it is HL-only content. Examiners expect to see the variables correctly separated, both integrals, the constant of integration, and clear use of any initial condition. Differentiate your final solution to verify that it satisfies the original differential equation.