JEE Main & Advanced Mathematics

Differential Equations

High Weightage • 2–4 Questions Expected

Basic Definitions

Differential Equation: A mathematical equation that relates some function with its derivatives.

Order and Degree of a Differential Equation

Order: Order of the highest differential coefficient appearing in the equation.

Degree: Exponent of the highest differential coefficient, when the differential equation is a polynomial in all differential coefficients.

Formation of Differential Equations

Consider a family of curves f(x, y, α₁, α₂, …, αₙ) = 0, where α₁, α₂, …, αₙ are n independent parameters.

First Order Differential Equations

1. Separable Variables

dy/dx = M(x)/N(y) ⇒ N(y) dy = M(x) dx
∫N(y) dy = ∫M(x) dx + c

2. Reducible to Separable Form

dy/dx = f(ax + by + c) → Put ax + by + c = t

3. Homogeneous Differential Equations

dy/dx = f(x,y) where f(x,y) = F(y/x) or F(x/y)

Put y = vx (v = v(x)) → dy/dx = v + x dv/dx

4. Equations Reducible to Homogeneous Form

dy/dx = (ax + by + c)/(Ax + By + C) (aB ≠ Ab)

Substitute x = X + h, y = Y + k where h, k chosen such that:

ah + bk + c = 0
Ah + Bk + C = 0

h = (bC − Bc)/(aB − Ab) , k = (Ac − aC)/(aB − Ab)

Then equation becomes homogeneous in X, Y → Put Y = VX

If aB = Ab → Put ax + by = t

5. First Order Linear Differential Equations

dy/dx + Py = Q (P, Q functions of x only)

Integrating Factor = e^{∫P dx}

y × e^{∫P dx} = ∫ Q e^{∫P dx} dx + c

General Forms of Variable Separation

FormEquivalent Differential
(i) d(x + y)dx + dy
(ii) d(xy)y dx + x dy
(iii) d(x/y)(y dx − x dy)/y²
(iv) d(y/x)(x dy − y dx)/x²
(v) d(log xy)(y dx + x dy)/xy
(vi) d(log(y/x))(x dy − y dx)/xy
(vii) d(log(x² + y²)/(x − y))2(x dy − y dx)/(x² − y²)
(viii) d(tan⁻¹(y/x))(x dy − y dx)/(x² + y²)
(ix) d[f(x,y)]^{1−n}/(1−n)[f(x,y)]^{-n} f'(x,y)
(x) d(√(x² + y²))(x dx + y dy)/√(x² + y²)

Orthogonal Trajectory

Any curve which cuts every member of a given family of curves at right angle.

Example: y = kx are orthogonal trajectories of the family x² + y² = a² (circles through origin).

Applications of Differential Equations

Geometrical Applications

Equation of tangent at (x,y) on y = f(x):

Y − y = (dy/dx)(X − x)

At X-axis (Y=0): X = x − y/(dy/dx)

At Y-axis (X=0): Y = y − x(dy/dx)

For normal: (Y − y)(dy/dx) + (X − x) = 0

JEE Main/Advanced Weightage

• 2–4 questions expected every year

• Must know: Separable, Homogeneous, Linear, Reducible forms, Orthogonal trajectories

• High probability: Equations of form dy/dx = f(ax+by+c), variable separable tricks, tangent/normal length problems