How to solve differential equations — classify type and pick method

Question

Given a differential equation, how do you classify it and choose the right solution method — variable separable, homogeneous, linear, or exact? Provide a decision framework.

(JEE Main tests DE type identification in 1-2 questions; CBSE 12 asks specific solution methods)

Solution — Step by Step

Can you write the equation as $f(y)\,dy = g(x)\,dx$ ? If yes, just integrate both sides.

Example: $\frac{dy}{dx} = \frac{x}{y}$ → $y\,dy = x\,dx$ → $\frac{y^2}{2} = \frac{x^2}{2} + C$

When to use: When all $y$ terms can be moved to one side and all $x$ terms to the other.

If $\frac{dy}{dx} = F\left(\frac{y}{x}\right)$ or every term has the same degree in $x$ and $y$ , the equation is homogeneous.

Substitute $y = vx$ (so $dy/dx = v + x\,dv/dx$ ). This converts it to a variable separable equation in $v$ and $x$ .

Example: $\frac{dy}{dx} = \frac{x + y}{x}$ → let $y = vx$ → becomes separable.

Standard form: $\frac{dy}{dx} + P(x)y = Q(x)$

This is solved using the integrating factor method:

\text{IF} = e^{\int P(x)\,dx}

Multiply throughout by IF, and the left side becomes $\frac{d}{dx}(y \cdot \text{IF})$ .

y \cdot \text{IF} = \int Q(x) \cdot \text{IF}\,dx + C

Bernoulli equation: $\frac{dy}{dx} + P(x)y = Q(x)y^n$ . Substitute $v = y^{1-n}$ to reduce it to linear form.

Exact equation: $M(x,y)\,dx + N(x,y)\,dy = 0$ where $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ . The solution is found by integration: $\int M\,dx + \int (\text{terms in } N \text{ not in } \partial/\partial y \int M\,dx)\,dy = C$ .

flowchart TD
    A["Given DE: dy/dx = ..."] --> B{Can separate x and y?}
    B -->|Yes| C["Variable Separable<br/>Integrate both sides"]
    B -->|No| D{Same degree in x and y?}
    D -->|Yes| E["Homogeneous<br/>Put y = vx"]
    D -->|No| F{Form: dy/dx + Py = Q?}
    F -->|Yes| G["Linear<br/>IF = e^∫P dx"]
    F -->|No| H{Form: dy/dx + Py = Qy^n?}
    H -->|Yes| I["Bernoulli<br/>Substitute v = y^(1−n)"]
    H -->|No| J["Check for Exact or<br/>use other methods"]

Why This Works

Each type of DE has a specific structure, and the solution method exploits that structure. Variable separable directly uses the fundamental theorem of calculus. Homogeneous equations have a scaling symmetry that the substitution $y = vx$ exploits. Linear DEs have a special property: the integrating factor makes the left side a perfect derivative.

The decision tree works from simplest to most complex. Always try variable separable first — it is the fastest. If that fails, check homogeneity, then linearity.

Alternative Method

For CBSE boards, 90% of questions are either variable separable or linear. If you master just these two methods thoroughly, you will handle most DE questions. The integrating factor for a linear DE can be computed quickly: if $P(x) = 2/x$ , then $\text{IF} = e^{\int 2/x\,dx} = e^{2\ln x} = x^2$ .

Common Mistake

Students try to apply the linear DE method to a non-linear equation. The method $\frac{dy}{dx} + Py = Q$ only works when the equation is linear in y — meaning $y$ appears only to the first power (no $y^2$ , no $\sin y$ , no $e^y$ ). If $y$ appears in a non-linear form, you need a different approach (Bernoulli substitution, or variable separable after rearrangement). Always verify linearity before computing the integrating factor.