Differential Equations

Concept overview

A differential equation relates a function to its derivatives. Order is the highest derivative appearing; degree is the power of the highest derivative after clearing radicals. First-order DEs dominate JEE Main.

Variables-separable form dy/dx = f(x) g(y) is solved by moving terms and integrating: ∫dy/g(y) = ∫f(x) dx. This is the simplest and most-tested case.

Linear first-order dy/dx + P(x) y = Q(x): multiply by integrating factor e^(∫P dx) to make the left side an exact derivative. The solution is y · e^(∫P dx) = ∫Q · e^(∫P dx) dx + C.

Exact DEs M dx + N dy = 0 where ∂M/∂y = ∂N/∂x have a solution F(x, y) = C where ∂F/∂x = M and ∂F/∂y = N. If not exact, sometimes an integrating factor makes it so.

Application: growth/decay dy/dt = ky gives y = y₀ eᵏᵗ (exponential model — radioactive decay, cooling, population). Newton's law of cooling and orthogonal trajectories are standard JEE application problems.

Key formulas & concepts

• Order and degree: Order = highest derivative; degree = power after clearing radicals
• Variables separable: dy/dx = f(x) g(y) ⇒ ∫dy/g(y) = ∫f(x) dx
• Linear first-order: dy/dx + P(x)y = Q(x)
• Integrating factor: μ(x) = e^(∫P(x) dx)
• Exact condition: ∂M/∂y = ∂N/∂x for M dx + N dy = 0
• Exponential model: dy/dt = ky ⇒ y = y₀ eᵏᵗ

Common mistakes to avoid

• Forgetting to include the integration constant C in the general solution.
• Missing the integrating factor step in linear DEs.
• Confusing order with degree — order counts the highest derivative; degree its power after clearing radicals.
• Solving a non-exact DE as if it were exact.

JEE Main relevance

JEE Main asks 1–2 differential equation problems every paper — usually one variables-separable or linear DE, and often one application (Newton's cooling, growth/decay).

JEE Advanced relevance

Advanced problems combine differential equations with initial-value conditions or ask about specific solution curves passing through given points.

Previous-year questions (PYQ)

Variables-separable is the top PYQ archetype. Drill 15 problems and this chapter is largely done.