Differential Equations for JEE Main: Separable & Linear

Differential equations contributes two to three questions every JEE Main session and is one of the more structured chapters in the Mathematics syllabus: the question types are well-defined, the solution methods are algorithmic, and recognising which method applies (variable separable, homogeneous, linear) is the main skill being tested. This guide covers all three solution methods in the order you should master them, with the exact problem features that trigger each method.

Order, Degree, and Formation

The order of a DE is the highest derivative present. The degree is the power of the highest-order derivative when the DE is written as a polynomial in derivatives (no square roots or fractional powers of derivatives). JEE tests order and degree identification directly, including cases where the degree is undefined (when the DE cannot be written as a polynomial — e.g., when the derivative appears inside a log or exponential). A degree-1, order-1 DE is linear if y and dy/dx both appear to the first power with no products between them.

Formation of a DE from a given family of curves: differentiate the curve equation once for each arbitrary constant and eliminate the constants. Two-constant family → second-order DE. Three-constant → third-order. This formation process is a reliable 3-mark question. The key is counting how many differentiations are needed (equals the number of arbitrary constants) and then eliminating the constants systematically. Take a free differential equations mock to test your formation and solution skills.

Variable Separable Method

A DE is separable if it can be written as f(x)dx = g(y)dy — all x terms on one side and all y terms on the other. Integrate both sides to get the general solution. Standard form to recognise: dy/dx = f(x)/g(y) → g(y)dy = f(x)dx. Variations that initially appear non-separable often become separable after substitution: if dy/dx = f(ax+by+c), substitute v = ax+by+c to get a separable DE in v and x.

JEE tests separable DEs most frequently through physical applications: exponential growth/decay (dN/dt = kN), Newton's cooling (dθ/dt = k(θ−θ₀)), and mixing problems (rate of change = rate in − rate out). In each case, write the rate equation, identify the separable form, integrate both sides, and apply the initial condition to find the particular solution.

Homogeneous and Linear First-Order DEs

A DE is homogeneous if it can be written as dy/dx = F(y/x). Substitute y = vx (so dy/dx = v + x·dv/dx) to convert it to a separable DE in v and x, then back-substitute. The indicator: every term in the DE has the same total degree in x and y (degree-n homogeneous if each term has degree n). JEE presents these by giving a DE like dy/dx = (y² − x²)/(2xy) and asking for the solution.

A linear first-order DE has the form dy/dx + P(x)y = Q(x). The integrating factor IF = e^(∫P dx). Multiply both sides by IF: d/dx(y·IF) = Q·IF. Integrate both sides. The IF method converts every linear first-order DE into a direct integration. JEE also tests the Bernoulli DE dy/dx + P(x)y = Q(x)yⁿ — substitute v = y^(1−n) to linearise. For the integration techniques needed to apply these methods, see our definite integration guide. For the broader calculus context, see our limits and continuity guide.