Class 12 Maths NCERT Solutions Chapter 9 Differential Equations | Boundless Maths

Unit III · Calculus · Chapter 9 NCERT Solutions Class 12 Maths

Class 12 Maths NCERT Solutions Chapter 9 Differential Equations

Free, step-by-step Class 12 Maths NCERT Solutions Chapter 9 Differential Equations, covering order and degree, general and particular solutions, and all three solving methods — variable separable, homogeneous, and linear differential equations. Every question across all five exercises and the Miscellaneous Exercise is solved the way CBSE awards marks, with the key formulas and a decision guide for choosing the right method, right on this page.

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Class 12 Maths NCERT Solutions Chapter 9 — Overview

Differential Equations takes the differentiation and integration you've already learned and turns them into a tool for describing how things change. An equation like dy/dx + y = sin x doesn't ask you to find a number — it asks you to find an entire function that makes the equation true. This chapter builds that skill from the ground up: first defining what order and degree even mean, then verifying that a given function actually solves an equation, and finally teaching you three concrete methods — variable separable, homogeneous, and linear — for solving first order, first degree equations yourself.

These Class 12 Maths NCERT Solutions Chapter 9 Differential Equations work through every one of the 98 questions across Exercises 9.1 to 9.5 and the Miscellaneous Exercise, including the word-based growth, decay, and geometry problems that show up almost every year in the board exam. Each solution is written the way CBSE's marking scheme expects — separating variables cleanly, stating the integrating factor explicitly, and never skipping the step of replacing the arbitrary constant using the given initial condition.

How the Chapter Builds

One Idea Leads to the Next

Order & Degree

Identify the highest derivative present, and whether the equation is a polynomial in its derivatives. Exercise 9.1.

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Verifying Solutions

Substitute a given function (and its derivatives) back into the equation to confirm it's a genuine solution. Exercise 9.2.

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Variable Separable

When x-terms and y-terms can be fully split apart, integrate each side on its own. Exercise 9.3.

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Homogeneous Equations

When the equation only depends on y/x, substitute y = vx to reduce it back to a separable form. Exercise 9.4.

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Linear Equations

For equations of the form dy/dx + Py = Q, multiply by an integrating factor to solve directly. Exercise 9.5.

Quick Reference

Important Formulas — Chapter 9

Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below have the full printable version for all of Calculus.

Order and Degree (§9.1)

Order

\text{Order} = \text{order of the highest derivative present}

Order and degree, when defined, are always positive integers.

Degree

\text{Degree} = \text{highest power of the highest-order derivative}

Only defined if the equation is a polynomial in its derivatives — an equation like y' + sin(y') = 0 has undefined degree.

General & Particular Solutions (§9.3)

General solution

Number of arbitrary constants = order of the differential equation

A particular solution is obtained by assigning specific values to those constants using given conditions.

Variable Separable Method (§9.3)

Separating the variables

\dfrac{dy}{dx} = h(y)\,g(x) \;\Rightarrow\; \int \dfrac{1}{h(y)}\,dy = \int g(x)\,dx

Works whenever F(x, y) can be written as a pure product of an x-function and a y-function.

Growth / decay model

\dfrac{dP}{dt} = kP \;\Rightarrow\; P = C\,e^{kt}

The standard set-up behind almost every "increases at the rate of r% per year" or bacteria-culture question in Ex 9.3.

Homogeneous Differential Equations (§9.4)

Homogeneous function of degree n

F(\lambda x, \lambda y) = \lambda^{n}\, F(x, y)

The differential equation dy/dx = F(x, y) is homogeneous when F is homogeneous of degree zero.

Standard substitution

y = vx \;\Rightarrow\; \dfrac{dy}{dx} = v + x\dfrac{dv}{dx}

Reduces the homogeneous equation back to a variable separable equation in v and x.

Alternative substitution

x = vy \;\Rightarrow\; \dfrac{dx}{dy} = v + y\dfrac{dv}{dy}

Used when the equation is more naturally written as dx/dy = F(x, y).

Linear Differential Equations (§9.5)

Standard form and Integrating Factor

\dfrac{dy}{dx} + Py = Q \;\Rightarrow\; \text{I.F.} = e^{\int P\,dx}

P and Q are constants or functions of x only.

Solution using the Integrating Factor

y \cdot (\text{I.F.}) = \int Q \cdot (\text{I.F.})\,dx + C

The I.F. is chosen precisely so the left side becomes the derivative of y · (I.F.).

The other linear form (in x)

\dfrac{dx}{dy} + P_1 x = Q_1 \;\Rightarrow\; \text{I.F.} = e^{\int P_1\,dy}

Use this form whenever P and Q would otherwise need to depend on y instead of x.

Decision Guide

Which Method Should I Use?

A quick way to decide once you've written the equation in the form dy/dx = F(x, y).

Situation	Use this	Why
F(x, y) can be written as a pure product h(y)·g(x)	Variable Separable	Fastest — split the variables and integrate each side directly, no substitution needed.
F(x, y) is homogeneous of degree zero, i.e. dy/dx depends only on y/x	Substitute y = vx (or x = vy)	Converts the equation into a new one in v and x that is variable separable.
The equation is linear in y — matches dy/dx + Py = Q, with P, Q functions of x only	Integrating Factor method	Multiplying by e^∫P dx turns the left side into an exact derivative you can integrate directly.
The question is a word problem — growth, decay, or a rate stated in words	Translate to a DE first, then usually Variable Separable	Most Ex 9.3 word problems reduce to dy/dx = ky once you write the rate condition as an equation.

Solve Chapter-Wise

Class 12 Maths NCERT Solutions Chapter 9 — Choose an Exercise

9.1

Exercise 9.1

Order and degree of differential equations — definitions and worked classification · 12 questions

Solve Exercise 9.1 →

9.2

Exercise 9.2

Verifying that a given function, explicit or implicit, solves the corresponding differential equation · 12 questions

Solve Exercise 9.2 →

9.3

Exercise 9.3

Variable separable method — general and particular solutions, curve equations, growth and decay applications · 23 questions

Solve Exercise 9.3 →

9.4

Exercise 9.4

Homogeneous differential equations — proving homogeneity and solving with the y = vx substitution · 17 questions

Solve Exercise 9.4 →

9.5

Exercise 9.5

Linear differential equations — the integrating factor method, in both dy/dx and dx/dy forms · 19 questions

Solve Exercise 9.5 →

Miscellaneous Exercise

Mixed questions combining all three methods, plus curve equations and particular solutions · 15 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

Every formula for Calculus — differentiation, applications of derivatives, integration, and differential equations — in one printable set of Formula Cards.

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Common Questions

Frequently Asked Questions

Quick answers about Chapter 9, Differential Equations.

How many exercises are there in Chapter 9, Differential Equations?

There are five exercises — 9.1 to 9.5 — plus a Miscellaneous Exercise, totalling 98 questions across order/degree, verifying solutions, the variable separable method, homogeneous equations, and linear equations.

What are the three methods used to solve first order, first degree differential equations in this chapter?

The variable separable method (Exercise 9.3), the homogeneous equation method using the substitution y = vx (Exercise 9.4), and the linear differential equation method using an integrating factor (Exercise 9.5).

What is the difference between the order and degree of a differential equation?

Order is the order of the highest derivative appearing in the equation. Degree is the highest power of that highest-order derivative — but only when the equation is a polynomial in its derivatives, otherwise the degree is not defined.

Is Differential Equations important for the CBSE board exam?

Yes — it's a core Calculus-unit chapter with near-guaranteed weightage every year, including at least one higher-mark question built around the variable separable, homogeneous, or linear method.

What should I revise before starting this chapter?

Make sure integration techniques from Chapter 7 (Integrals) are solid, since every method in this chapter ends with integrating both sides of an equation — substitution, partial fractions, and standard integrals all show up repeatedly.

Where can I find the official NCERT textbook for this chapter?

The official NCERT Class 12 Maths textbook, including Chapter 9 on Differential Equations, is available for free on the NCERT website.