NCERT Solutions for Class 12 Mathematics Chapter 9: Differential Equations – Free PDF Download

Chapter 9 of Class 12 Maths is Differential Equations. It covers the order and degree of differential equations, forming them, and solving first-order first-degree equations by variable separation, homogeneous methods, and linear equations. These Class 12 Mathematics Chapter 9 solutions are also useful as quick revision notes before exams.

Basic Definitions

A differential equation involves derivatives of a dependent variable with respect to an independent variable. The order is the highest derivative present; the degree is the power of the highest-order derivative (when the equation is a polynomial in derivatives).

Formation of Differential Equations

Given a family of curves with n arbitrary constants, differentiating n times and eliminating the constants gives a differential equation of order n.

Variable Separable Method

If the equation can be written as f(y)dy=g(x)dx, integrate both sides separately: ∫f(y)dy=∫g(x)dx+C.

Homogeneous Differential Equations

An equation dy/dx=f(x,y) is homogeneous if f(λx,λy)=f(x,y). Solved by substituting y=vx (so dy/dx=v+x dv/dx), converting to a variable-separable form in v and x.

Linear Differential Equations

A linear equation has the form dy/dx+Py=Q (P,Q functions of x only). Solved using the integrating factor IF=e^{∫P dx}; the solution is y·IF=∫(Q·IF)dx+C.

Frequently Asked Questions

What is the order and degree of a differential equation?
Order is the highest derivative present; degree is the power of that highest derivative (equation must be polynomial in derivatives).

How do you solve a homogeneous differential equation?
Substitute y=vx to convert to a variable-separable equation in v and x.

What is the integrating factor for dy/dx+Py=Q?
IF=e^{∫P dx}.

Written by Satish

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