JEE Main Maths Differential Equations 2027: Order, Degree, Separable, Homogeneous & Linear

The JEE Main syllabus keeps this chapter tight: only three solving methods are named. That is good news, because it means every question is really asking you to spot which of the three types you are looking at, and then apply a fixed recipe.

Before solving a differential equation, ask which of the three recipes it fits. The recipe decides everything that follows.

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Chapter at a Glance

Priority reflects past-paper trends, not an official NTA weightage.

What the NTA Syllabus Covers

• Ordinary differential equations, their order and degree
• Solution of a differential equation by the method of separation of variables
• Solution of a homogeneous differential equation
• Solution of a linear differential equation of the type $\dfrac{dy}{dx} + p(x)y = q(x)$

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Master These Topics

1. Order and Degree

The order is the order of the highest derivative present. The degree is the power of the highest-order derivative, once the equation is written as a polynomial in derivatives (free of roots and fractional powers of derivatives).

Worked example: $\left(\dfrac{d^2y}{dx^2}\right)^3 + \left(\dfrac{dy}{dx}\right)^2 + y = 0$ has order 2 (from $d^2y/dx^2$) and degree 3 (its power).

Trap: If the equation contains a derivative inside a square root or a trigonometric function, it must first be rearranged into polynomial form before the degree can be read off.

2. Separation of Variables

If the equation can be written as $\dfrac{dy}{dx} = f(x)g(y)$, separate the variables and integrate both sides.

Worked example: $\dfrac{dy}{dx} = y$. Separating gives $\dfrac{dy}{y} = dx$, so $\ln\lvert y \rvert = x + C$, giving $y = Ae^x$. If the curve passes through $(0, 1)$, then $A = 1$ and $y = e^x$.

3. Homogeneous Differential Equations

A first-order equation is homogeneous if it can be written as $\dfrac{dy}{dx} = f\!\left(\dfrac{y}{x}\right)$. Substitute $y = vx$, so $\dfrac{dy}{dx} = v + x\dfrac{dv}{dx}$, which turns the equation into a separable one in $v$ and $x$.

Worked example: $\dfrac{dy}{dx} = \dfrac{x^2 + y^2}{xy}$. Dividing through by $x^2$ gives $\dfrac{dy}{dx} = \dfrac{1 + (y/x)^2}{y/x}$. With $y = vx$, this becomes $v + x\dfrac{dv}{dx} = \dfrac{1 + v^2}{v}$, so $x\dfrac{dv}{dx} = \dfrac{1}{v}$. Separating and integrating gives $v^2 = 2\ln\lvert x \rvert + C$, and substituting back $v = y/x$ gives the solution in terms of $x$ and $y$.

4. Linear Differential Equations

For $\dfrac{dy}{dx} + P(x)y = Q(x)$, the integrating factor is

$$\text{I.F.} = e^{\int P(x)\,dx}$$

and the solution is $y \cdot \text{I.F.} = \displaystyle\int Q(x) \cdot \text{I.F.}\,dx + C$.

Worked example: $\dfrac{dy}{dx} + \dfrac{y}{x} = x$. Here $P = 1/x$, so $\text{I.F.} = e^{\int dx/x} = e^{\ln x} = x$. Then

$$xy = \int x \cdot x \,dx = \frac{x^3}{3} + C \quad \Longrightarrow \quad y = \frac{x^2}{3} + \frac{C}{x}$$

Trap: The equation must be in the exact form $\dfrac{dy}{dx} + P(x)y = Q(x)$, with the coefficient of $\dfrac{dy}{dx}$ equal to 1, before reading off $P(x)$.

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Common Traps to Avoid

• Reading off the degree before clearing radicals or fractional powers of derivatives.
• Forgetting the substitution $\dfrac{dy}{dx} = v + x\dfrac{dv}{dx}$ when solving a homogeneous equation.
• Using the wrong sign or forgetting a constant when finding the integrating factor.
• Not dividing through to make the coefficient of $dy/dx$ equal to 1 before identifying $P(x)$.

60-Second Revision Sheet

• Order: highest derivative present; degree: its power once polynomial in derivatives
• Separable: $\dfrac{dy}{dx} = f(x)g(y)$, separate and integrate
• Homogeneous: substitute $y = vx$
• Linear: $\text{I.F.} = e^{\int P\,dx}$, solution $y \cdot \text{I.F.} = \int Q \cdot \text{I.F.}\,dx + C$

Your Study Plan

1. Day 1: order, degree and separable equations.
2. Day 2: homogeneous equations with the $v = y/x$ substitution.
3. Day 3: linear equations and integrating factor practice.
4. Day 4: mixed identification (which type is it?) and a timed set.

Practice Differential Equations Questions Free → (opens in a new tab)

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Continue Your Maths Journey

• Previous chapter: Integral Calculus
• Next chapter: Co-ordinate Geometry
• All 14 JEE Main Maths chapters
• Complete JEE Main Syllabus 2027 guide

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