Differential Equations

part 1

Basic Concepts

• Definition — differential equations
• Order and degree of differential equations
• General and particular solutions
• Formation of differential equations

part 2

Methods of Solution

• Variables separable method
• Homogeneous differential equations
• Solving homogeneous equations by substitution

part 3

Linear Differential Equations

• Linear differential equations of first order
• Integrating factor method
• Applications of differential equations
• Exponential growth and decay models

Variable separable method

$f(y)\,dy = g(x)\,dx \Rightarrow \int f(y)\,dy = \int g(x)\,dx + C$

★ board favourite

Homogeneous DE: substitute y = vx

$\frac{dy}{dx} = F\!\left(\frac{y}{x}\right)$ → y = vx → separate variables

★ board favourite

Linear DE: dy/dx + Py = Q

$\text{IF} = e^{\int P\,dx}$; solution: $y \cdot \text{IF} = \int Q \cdot \text{IF}\,dx + C$

★ board favourite

Linear DE in x: dx/dy + Px = Q

$\text{IF} = e^{\int P\,dy}$; solution: $x \cdot \text{IF} = \int Q \cdot \text{IF}\,dy + C$

★ frequently asked

Solve the differential equation: (x² + xy) dy = (x² + y²) dx.

Solve the differential equation: x (dy/dx) + 2y = x² log x.

Solve the differential equation: (x + y) dy/dx = 1. Also solve: dy/dx + 2y = e^x.

Find the general solution of the differential equation: dy/dx + y·cot x = 2x + x²·cot x, given x ≠ 0.

Solve the differential equation: dy/dx = (1 + y²)/(1 + x²). Find the particular solution when y = 1 at x = 0.

Show that the differential equation [x·sin²(y/x) − y] dx + x dy = 0 is homogeneous and solve it, given that y = π/4 when x = 1.

Find the order and degree (if defined) of the differential equation: (d²y/dx²)² + cos(dy/dx) = 0.

Find the particular solution of the differential equation: dy/dx = 1 + x + y + xy, given that y = 0 when x = 1.

Find the integrating factor of the differential equation: x (dy/dx) − y = 2x².

Solve the differential equation: (1 + x²) dy/dx + 2xy = 1/(1 + x²), given that y = 0 when x = 1.

How many marks is Differential Equations worth in the Class 12 board exam?

Around 6 marks of the 80-mark CBSE Maths theory paper, based on the official unit-wise weightage. Linear DE with integrating factor — usually a 5-marker.

Is Differential Equations easy or hard?

It's rated hard — one of the tougher chapters in Class 12 Maths. Most students need about 4 hours to cover it well.

What are the important topics in Differential Equations?

The chapter covers 11 NCERT topics in 3 areas: Basic Concepts; Methods of Solution; Linear Differential Equations.

What questions come from Differential Equations in board exams?

Between 2018–2024, CBSE board papers asked questions from this chapter on Homogeneous Differential Equations, Linear Differential Equations, Variable Separable, Order and Degree — 1- to 5-mark questions.