Differential EquationsDifferential EquationsCollege

Separable Equations

Separate variables, integrate both sides and apply the initial condition.

A differential equation relates a function to its derivatives. It is separable when it can be written $\frac{dy}{dx} = g(x)h(y)$ , so that all the $y$ terms can be moved to one side and all the $x$ terms to the other: $\frac{dy}{h(y)} = g(x)\,dx$ . Integrating both sides then gives the solution, with one constant of integration.

Separable equations model exponential growth and decay ( $\frac{dy}{dt} = ky$ ), Newton's law of cooling and logistic population growth. An initial condition such as $y(0) = 5$ fixes the constant and picks out one particular solution from the family.

Key terms

Separable form: The right side factors into a function of x times a function of y.
$\frac{dy}{dx} = g(x)h(y)$

Key ideas

Divide by $h(y)$ only where $h(y) \ne 0$ ; constant solutions $h(y) = 0$ may need to be added back.
One constant $C$ is enough after integrating both sides.
Solve for $y$ explicitly when possible, then apply the initial condition.
$\frac{dy}{dt} = ky$ has solution $y = y_0 e^{kt}$ .

Worked example

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

Separate: $\frac{dy}{y} = 2x\,dx$ .
Integrate: $\ln|y| = x^2 + C$ .
Exponentiate: $y = Ae^{x^2}$ .
Initial condition: $3 = A$ , so $y = 3e^{x^2}$ .

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

Common mistakes

Adding a constant to each side and keeping two constants.
Forgetting the absolute value inside the logarithm.
Applying the initial condition before integrating.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.For $y = 3e^{x^2}$ , find $y(1)$ to two decimals.
Show answer
$8.15$
2.A quantity obeys $\frac{dy}{dt} = 0.1y$ with $y(0) = 50$ . Find $y(10)$ to one decimal.
Show answer
$135.9$
3.A substance decays with $k = -\ln 2 / 5$ . What fraction remains after 10 years?
Show answer
$\frac{1}{4}$

Key terms

Key ideas

Worked example

Common mistakes

Practice questions

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