1. First-Order Separable

Video coming soon2 worked examples

Theory

A first-order differential equation is separable if it can be written as $\frac{dy}{dx} = f(x)g(y)$ . Separate the variables so that all the $y$ terms (with $dy$ ) are on one side and all the $x$ terms (with $dx$ ) on the other, then integrate both sides.

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

The general solution contains an arbitrary constant; a particular solution uses an initial condition to fix it.

The Golden Rule: separate first, integrate both sides, and include just one constant of integration — then apply any initial condition.

⚠️ Common Examiner Traps

Two constants: only one constant of integration is needed — combine them into one.
Forgetting the condition: use the given initial condition to find the constant for a particular solution.
Separation slips: make sure every $y$ moves with $dy$ and every $x$ with $dx$ .

Worked examples

Example 1

Find the general solution of $\dfrac{dy}{dx} = xy$ .

Step 1: Separate the variables:

\frac{1}{y}\,dy = x\,dx

Step 2: Integrate both sides:

\ln|y| = \frac{x^2}{2} + c

Step 3: Exponentiate to make $y$ the subject:

y = A e^{x^2/2} \quad (A = e^c)

Example 2

Solve $\dfrac{dy}{dx} = 2xy^2$ given that $y = 1$ when $x = 0$ .

Step 1: Separate and integrate:

\int \frac{1}{y^2}\,dy = \int 2x\,dx \implies -\frac{1}{y} = x^2 + c

Step 2: Apply the condition $y=1$ at $x=0$ :

-\frac{1}{1} = 0 + c \implies c = -1

Step 3: Rearrange for $y$ :

-\frac{1}{y} = x^2 - 1 \implies y = \frac{1}{1 - x^2}