The Method of Separation of Variables
Definition
A first-order DE is called separable if it can be written in the form:
Separable Form
$$\frac{dy}{dx} = f(x) \cdot g(y)$$
Here, the right-hand side is a product of a function of $x$ alone and a function of $y$ alone.
The idea: separate variables to each side, then integrate both sides independently.
Steps to Solve
- Rewrite the DE in the form $\dfrac{dy}{dx} = f(x)\cdot g(y)$.
- Separate: $\dfrac{dy}{g(y)} = f(x)\,dx$ (assuming $g(y) \neq 0$).
- Integrate both sides: $\displaystyle\int \frac{dy}{g(y)} = \int f(x)\,dx + C$
- Simplify and write the general solution.
- If initial conditions are given, find the particular solution.
Worked Examples
Example 1
Solve: $\dfrac{dy}{dx} = \dfrac{x}{y}$
Separate: $y\,dy = x\,dx$
Integrate: $\displaystyle\int y\,dy = \int x\,dx$
$$\frac{y^2}{2} = \frac{x^2}{2} + C \implies \boxed{y^2 - x^2 = K}$$
where $K = 2C$ is an arbitrary constant. This is a family of rectangular hyperbolas.
Example 2
Solve: $\dfrac{dy}{dx} = e^{x+y}$
Rewrite: $\dfrac{dy}{dx} = e^x \cdot e^y$
Separate: $e^{-y}\,dy = e^x\,dx$
Integrate: $-e^{-y} = e^x + C$
$$\boxed{e^x + e^{-y} = C}$$
Example 3 — With Initial Condition
Solve $\dfrac{dy}{dx} = y^2$, given $y(0) = 1$
Separate: $\dfrac{dy}{y^2} = dx$
Integrate: $-\dfrac{1}{y} = x + C$
Apply $y(0)=1$: $-1 = 0 + C \Rightarrow C = -1$
$$-\frac{1}{y} = x - 1 \implies \boxed{y = \frac{1}{1-x}}$$
Example 4
Solve: $(1+x^2)\,dy = (1+y^2)\,dx$
Separate: $\dfrac{dy}{1+y^2} = \dfrac{dx}{1+x^2}$
Integrate both sides:
$$\tan^{-1}y = \tan^{-1}x + C \implies \boxed{\tan^{-1}y - \tan^{-1}x = C}$$
Common Mistakes to Avoid
| Mistake | Correct Approach |
|---|---|
| Forgetting the constant of integration $C$ | Always add $+ C$ on one side after integrating |
| Dividing by $g(y) = 0$ without checking | Separately check if $g(y) = 0$ gives a solution |
| Adding $C$ on both sides | Combine: $C_1 - C_2 = C$ (one constant suffices) |
🔑 Key Takeaways
- Use this method when DE can be written as $\frac{dy}{dx} = f(x)\cdot g(y)$.
- Bring all $y$-terms to left and all $x$-terms to right, then integrate.
- Always add the arbitrary constant $C$.
- Apply initial conditions (if given) to find the particular solution.
- Always check if $g(y) = 0$ is also a valid (trivial) solution.