PYQ (JEE Main & Advanced)

High-Yield JEE Main Concepts for Differential Equations

1. Exact Differentials (Method of Inspection)

Many JEE Main and JEE Advanced problems become much shorter if we recognize standard exact differentials directly.

• $x \, dy + y \, dx = d(xy)$
• $\dfrac{x \, dy - y \, dx}{x^2} = d\left(\dfrac{y}{x}\right)$
• $\dfrac{y \, dx - x \, dy}{y^2} = d\left(\dfrac{x}{y}\right)$
• $\dfrac{x \, dx + y \, dy}{x^2+y^2} = \dfrac{1}{2} d\big(\ln(x^2+y^2)\big)$
• $\dfrac{x \, dy - y \, dx}{x^2+y^2} = d\left(\tan^{-1}\dfrac{y}{x}\right)$

These patterns are extremely useful in objective questions because they avoid a long derivation.

2. Orthogonal Trajectories

If one family of curves cuts another at right angles, then their slopes are negative reciprocals of each other. To find orthogonal trajectories:

1. Start from the given family of curves.
2. Differentiate and eliminate the arbitrary constant to get its differential equation.
3. Replace $\dfrac{dy}{dx}$ by $-\dfrac{dx}{dy}$, or equivalently replace slope $m$ by $-1/m$.
4. Solve the resulting differential equation.

3. Bernoulli's Equation

An equation of the form $$\frac{dy}{dx} + Py = Qy^n$$ is called a Bernoulli equation. It is not linear in $y$, but after dividing by $y^n$ and substituting $$v = y^{1-n},$$ it becomes a linear differential equation in $v$. This is a very standard JEE transformation.

4. Population Growth & Radioactive Decay

These are direct applications of separable differential equations.

• Exponential growth: $$\frac{dP}{dt} = kP \implies P = P_0 e^{kt}$$
• Exponential decay: $$\frac{dN}{dt} = -\lambda N \implies N = N_0 e^{-\lambda t}$$ Such models are based on the idea that the rate of change is proportional to the quantity present.

5. Substitutions for Special Forms

If a differential equation contains a linear expression such as $ax+by+c$, then putting $$v=ax+by+c$$ often simplifies the equation into separable form. This is especially common in equations like $$\frac{dy}{dx}=f(ax+by+c).$$