Summary
NCERT Class 12 Mathematics Chapter 9 covers Differential Equations, teaching students how to form and solve first-order differential equations using three methods: variables separable, homogeneous, and linear differential equations.
Chapter 9 of NCERT Class 12 Mathematics Part II introduces differential equations — equations involving derivatives of a dependent variable with respect to an independent variable. The chapter covers key concepts including order (highest order derivative present) and degree (highest power of the highest order derivative, when the equation is polynomial in derivatives). Students learn to distinguish general solutions (containing arbitrary constants) from particular solutions, and to solve first-order first-degree differential equations using three methods: separation of variables, the substitution y = vx for homogeneous equations, and the integrating factor method for linear equations of the form dy/dx + Py = Q. Applications include population growth, compound interest, and curve geometry.
Key points & formulas
- 01Order of a differential equation is defined as the order of the highest order derivative of the dependent variable present in the equation; order and degree are always positive integers when defined.
- 02Degree is the highest power of the highest order derivative, but only when the equation is a polynomial in its derivatives; equations like y′′′ + y² + e^(y′) = 0 have undefined degree.
- 03The general solution contains as many arbitrary constants as the order of the equation; a particular solution is obtained by assigning specific values to those constants.
- 04Variables separable method rewrites dy/dx = h(y)·g(x) as (1/h(y))dy = g(x)dx and integrates both sides independently.
- 05A homogeneous differential equation has F(x,y) as a homogeneous function of degree zero; it is solved by substituting y = vx (or x = vy), which reduces it to a separable equation.
- 06A first-order linear differential equation dy/dx + Py = Q is solved by multiplying through by the integrating factor e^(∫P dx) and integrating both sides.
Frequently asked questions
01What is the difference between the order and degree of a differential equation?
Order is the order of the highest order derivative present in the equation. Degree is the highest power of that highest order derivative, but only when the equation is a polynomial in its derivatives — if the equation involves terms like sin(y′) or e^(y′), the degree is not defined.
02How do you solve a homogeneous differential equation?
First verify that dy/dx = F(x,y) where F(x,y) is a homogeneous function of degree zero, meaning F(λx, λy) = λ⁰ F(x,y). Then substitute y = vx so that dy/dx = v + x(dv/dx), which converts the equation into a separable form in v and x that can be integrated directly.
03What is an integrating factor and when is it used?
An integrating factor (I.F.) is a function e^(∫P dx) used to solve first-order linear differential equations of the form dy/dx + Py = Q, where P and Q are functions of x only. Multiplying both sides by the I.F. makes the left-hand side an exact derivative, allowing direct integration to find the general solution.
04Is the NCERT Class 12 Maths Chapter 9 PDF free to download?
Yes, the NCERT Class 12 Mathematics Chapter 9 (Differential Equations) PDF is free to download on cbseprepmaster.com.
More chapters in Mathematics Part II
This is the complete Mathematics Part II Chapter 9 as published by NCERT — every diagram, solved example, and exercise included, free. Browse all CBSE Class 12 textbooks.
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