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Mathematics Grade XII

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Chapter 9: Differential Equations

Class 12 • Mathematics • Chapter 9

Differential Equations

Equations that involve a derivative, and the families of curves that solve them.

CBSE / NCERT Aligned

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Chapter Roadmap

Order and Degree • General and Particular Solutions • Forming a Differential Equation • Variable Separable Method • Homogeneous Equations • Linear Equations and the Integrating Factor

Equations About Change

Most laws of nature are stated not as fixed values but as rates: how fast a population grows, how quickly a hot drink cools, how charge flows in a circuit. An equation that links a function to its own derivatives is a differential equation, and solving one means finding the function whose rate of change behaves as the equation demands.

Because adding a constant does not change a derivative, a differential equation usually has a whole family of solutions, one curve for each value of the constant. Pinning down a single member needs an extra condition. This chapter classifies differential equations by order and degree, then solves the three standard types met at this level: variable separable, homogeneous and linear.

Foundation

The order of a differential equation is the highest derivative present; the degree is the power of that highest derivative once the equation is polynomial in derivatives. A general solution contains arbitrary constants; a particular solution fixes them using given conditions.

Key Terms You Must Know

Term

Meaning

Example

Differential equation

An equation involving derivatives of an unknown function.

dy/dx = 2x

Order

The highest derivative that appears.

y″ + y = 0 has order 2

Degree

The power of the highest derivative (when polynomial).

usually 1 at this level

General solution

A solution containing arbitrary constants.

y = x² + C

Particular solution

One curve found by fixing the constants.

y = x² + 1 when y(0) = 1

Integrating factor

A multiplier that makes a linear equation exact.

IF = e^(∫ P dx)

Core Concepts, Step by Step

1. Order and Degree

Reading off the order is easy: it is the highest derivative that appears. The degree is the power of that highest derivative, but only after the equation has been cleared of roots and fractions so that it is a polynomial in the derivatives. Most board equations turn out to have degree one. These two numbers classify an equation before any solving begins.

2. General and Particular Solutions

Solving a differential equation produces a general solution carrying as many arbitrary constants as the order. Each choice of constants gives one curve in the solution family. An initial or boundary condition, such as a known value at a point, fixes the constants and selects the single particular solution that fits the situation.

The general solution of dy/dx = 2x is a whole family of parabolas y = x² + C

Several parabolas y equals x squared plus C stacked vertically, the solution family of a differential equation

3. Forming a Differential Equation

We can run the process in reverse: starting from a family of curves with arbitrary constants, we differentiate enough times to eliminate the constants. What remains is the differential equation that the family satisfies. For example, the lines through the origin y = mx give, after differentiating, the equation dy/dx = y/x.

4. Variable Separable Equations

If a first-order equation can be rearranged so that all the y terms (with dy) sit on one side and all the x terms (with dx) on the other, it is separable. We then integrate both sides independently and combine the constants into one. This is the simplest and most common solving method.

5. Homogeneous Equations

When dy/dx can be written purely in terms of the ratio y/x, the equation is homogeneous. The substitution y = vx (so dy/dx = v + x dv/dx) turns it into a separable equation in v and x, which we solve and then change back. This handles many equations that are not directly separable.

6. Linear Equations and the Integrating Factor

A first-order linear equation has the form dy/dx + P(x) y = Q(x). Multiplying through by the integrating factor IF = e raised to the integral of P makes the left side the exact derivative of (IF × y). Integrating both sides then gives y directly. This single technique solves a large and important class of equations.

Key Results with Proofs

Result 1: Order Equals the Highest Derivative

Statement. The order of a differential equation is the order of the highest derivative occurring in it.

Proof

Order is a direct reading of the equation, not a calculation.

List every derivative that appears: dy/dx, d²y/dx², and so on. – scan the equation

The largest of these is the highest derivative present. – identify the top one

Its order is by definition the order of the equation. – read off the order

For y″ + 3y′ + 2y = 0 the highest is y″, so the order is 2. – worked instance

Degree is then the power of that highest derivative once the equation is polynomial in its derivatives.

Result 2: The Variable Separable Method

Statement. If dy/dx = f(x) g(y), then ∫ dy/g(y) = ∫ f(x) dx.

Proof

Separating the variables lets each side be integrated on its own.

Start from dy/dx = f(x) g(y). – the separable form

Divide by g(y) and multiply by dx: dy/g(y) = f(x) dx. – separate the variables

Integrate each side with respect to its own variable. – integrate both sides

∫ dy/g(y) = ∫ f(x) dx + C gives the general solution. – one combined constant

Always combine the two constants of integration into a single C.

Result 3: The Integrating Factor for Linear Equations

Statement. For dy/dx + P y = Q, the integrating factor IF = e^(∫ P dx) makes d/dx(IF · y) = IF · Q.

Proof

The integrating factor is built so the left side collapses into a single derivative.

Multiply the equation by IF = e^(∫ P dx). – multiply through

The left side becomes IF dy/dx + IF P y, which is exactly d/dx(IF · y) since (IF)′ = IF · P. – recognise the product derivative

So d/dx(IF · y) = IF · Q. – exact form

Integrate both sides and divide by IF to find y. – solve for y

This is why IF is chosen as e raised to the integral of the coefficient P.

Worked Examples

Example 1

Question: State the order and degree of d²y/dx² + 3(dy/dx)² + y = 0.

▶ Show full working

Find the highest derivative and its power.

Highest derivative is d²y/dx², so the order is 2. – order

That highest derivative appears to the first power, so the degree is 1. – degree

Answer: Order 2, degree 1.

Example 2

Question: State the order and degree of (dy/dx)³ + y = x.

▶ Show full working

Only a first derivative appears.

Highest derivative is dy/dx, so the order is 1. – order

It is raised to the power 3, so the degree is 3. – degree

Answer: Order 1, degree 3.

Example 3

Question: Solve dy/dx = 2x.

▶ Show full working

Integrate both sides.

y = ∫ 2x dx. – integrate

= x² + C. – general solution

Answer: y = x² + C.

Example 4

Question: Solve dy/dx = y.

▶ Show full working

Separate the variables.

dy/y = dx, then integrate: ln|y| = x + c. – separate and integrate

Exponentiate: y = C eₓ. – solve for y

Answer: y = C eₓ.

Example 5

Question: Solve dy/dx = x/y.

▶ Show full working

Variable separable.

y dy = x dx; integrate both sides. – separate

y²/2 = x²/2 + c, so y² = x² + C. – tidy the constant

Answer: y² = x² + C.

Example 6

Question: Solve dy/dx = (1 + y²)/(1 + x²).

▶ Show full working

Separate and use the arctan integral.

dy/(1 + y²) = dx/(1 + x²). – separate

Integrate: tan⁻¹y = tan⁻¹x + C. – standard integrals

Answer: tan⁻¹y = tan⁻¹x + C.

Example 7

Question: Solve the linear equation dy/dx + y = eₓ.

▶ Show full working

Use the integrating factor.

Here P = 1, so IF = e^(∫ 1 dx) = eₓ. – integrating factor

d/dx(eₓ y) = eₓ · eₓ = e^(2x); integrate to eₓ y = e^(2x)/2 + C. – integrate

Divide by eₓ: y = eₓ/2 + C e⁻ₓ. – solve for y

Answer: y = eₓ/2 + C e⁻ₓ.

Example 8

Question: Solve dy/dx + (1/x) y = x.

▶ Show full working

Linear with IF.

P = 1/x, so IF = e^(∫ dx/x) = e^(ln x) = x. – integrating factor

d/dx(x y) = x · x = x²; integrate to x y = x³/3 + C. – integrate

Divide by x: y = x²/3 + C/x. – solve for y

Answer: y = x²/3 + C/x.

Example 9

Question: Find the particular solution of dy/dx = 2x with y(0) = 1.

▶ Show full working

Find the general solution, then use the condition.

General solution: y = x² + C. – integrate

At x = 0, y = 1 gives 1 = 0 + C, so C = 1. – apply the condition

Answer: y = x² + 1.

Example 10

Question: Solve the homogeneous equation dy/dx = (x + y)/x.

▶ Show full working

Substitute y = vx.

Write the right side as 1 + y/x. Put y = vx, so dy/dx = v + x dv/dx. – substitution

v + x dv/dx = 1 + v gives x dv/dx = 1, so dv = dx/x. – separate

Integrate: v = ln|x| + C; since v = y/x, y = x(ln|x| + C). – back-substitute

Answer: y = x(ln|x| + C).

Example 11

Question: Form the differential equation of the family y = mx.

▶ Show full working

Differentiate to remove m.

dy/dx = m. – differentiate

Since m = y/x, substitute: dy/dx = y/x. – eliminate the constant

Answer: dy/dx = y/x.

Example 12

Question: Form the differential equation of the family y = A eₓ.

▶ Show full working

Differentiate to remove A.

dy/dx = A eₓ = y. – differentiate and recognise A eₓ = y

Answer: dy/dx = y.

Where You Meet This in Real Life

Population biology

Growth and decay of populations follow dy/dt proportional to y, giving exponential models.

Physics of cooling

Newton’s law of cooling is a differential equation linking temperature change to the temperature gap.

Electrical circuits

The current in a circuit with resistance and inductance obeys a first-order linear differential equation.

Pharmacology

Drug elimination from the bloodstream is modelled by a decay equation that sets safe dosing intervals.

Finance

Continuously compounded interest grows according to dy/dt = ry, a separable differential equation.

Practice Sets A–D

Practice Set A – Basics

A1. State the order of d³y/dx³ + y = 0.

▶ Reveal full working

Highest derivative.

Order 3.

Answer: 3.

A2. Solve dy/dx = 3x².

▶ Reveal full working

Integrate.

y = x³ + C.

Answer: y = x³ + C.

A3. Solve dy/dx = 5.

▶ Reveal full working

Integrate.

y = 5x + C.

Answer: y = 5x + C.

A4. What kind of solution contains arbitrary constants?

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Term.

The general solution.

Answer: General solution.

Practice Set B – Conceptual

B1. Why does a first-order equation have one arbitrary constant?

▶ Reveal full working

Link to integration.

Solving involves one integration, which introduces one constant. – one integration

So the general solution carries exactly one arbitrary constant. – one constant

Answer: Because one integration introduces one constant.

B2. How is the degree of a differential equation defined?

▶ Reveal full working

Power of the highest derivative.

It is the power of the highest-order derivative once the equation is polynomial in derivatives.

Answer: The power of the highest derivative (in polynomial form).

B3. Which method suits dy/dx = f(x) g(y)?

▶ Reveal full working

Name it.

The variable separable method.

Answer: Variable separable.

B4. What is the integrating factor for dy/dx + P y = Q?

▶ Reveal full working

Formula.

IF = e raised to the integral of P with respect to x.

Answer: e^(∫ P dx).

Practice Set C – Application / Numerical

C1. Solve dy/dx = eₓ.

▶ Reveal full working

Integrate.

y = ∫ eₓ dx = eₓ + C.

Answer: y = eₓ + C.

C2. Solve dy/dx = sec² x.

▶ Reveal full working

Integrate.

y = tan x + C.

Answer: y = tan x + C.

C3. Solve dy/dx = xy by separation.

▶ Reveal full working

Separate.

dy/y = x dx; integrate: ln|y| = x²/2 + c. – separate and integrate

y = C e^(x²/2). – solve for y

Answer: y = C e^(x²/2).

C4. Find the particular solution of dy/dx = 3x² with y(1) = 2.

▶ Reveal full working

General then condition.

y = x³ + C; at x = 1, 2 = 1 + C so C = 1. – apply condition

Answer: y = x³ + 1.

Practice Set D – HOTS / Multi-step

D1. Solve the linear equation dy/dx + 2y = 6 and find the particular solution with y(0) = 0.

▶ Reveal full working

IF then condition.

P = 2, IF = e^(2x); d/dx(e^(2x) y) = 6 e^(2x). – integrating factor

Integrate: e^(2x) y = 3 e^(2x) + C, so y = 3 + C e^(−2x). – general solution

At x = 0, y = 0 gives 0 = 3 + C, so C = −3; y = 3(1 − e^(−2x)). – particular solution

Answer: y = 3(1 − e^(−2x)).

D2. Solve dy/dx = (1 + y²)/(1 + x²) with y(0) = 1.

▶ Reveal full working

Separable, then apply the condition.

tan⁻¹y = tan⁻¹x + C. – general solution

At x = 0, y = 1: tan⁻¹1 = 0 + C, so C = π/4. – find C

tan⁻¹y = tan⁻¹x + π/4. – particular solution

Answer: tan⁻¹y = tan⁻¹x + π/4.

D3. Form the differential equation of the family y = A x².

▶ Reveal full working

Differentiate to eliminate A.

dy/dx = 2Ax; and A = y/x² from the family. – differentiate and isolate A

Substitute: dy/dx = 2(y/x²)x = 2y/x. – eliminate A

Answer: dy/dx = 2y/x.

D4. Solve the homogeneous equation dy/dx = (x² + y²)/(xy).

▶ Reveal full working

Use y = vx.

Right side = (1 + v²)/v with y = vx; then v + x dv/dx = (1 + v²)/v. – substitute

x dv/dx = 1/v, so v dv = dx/x; integrate: v²/2 = ln|x| + C. – separate and integrate

Since v = y/x: y²/(2x²) = ln|x| + C, i.e. y² = 2x²(ln|x| + C). – back-substitute

Answer: y² = 2x²(ln|x| + C).

Chapter Summary

Everything in One Glance

Order & Degree

Order is the highest derivative; degree is its power once polynomial in derivatives.

Solutions

General solutions carry arbitrary constants; conditions fix them to give a particular solution.

Forming Equations

Differentiate a family enough times to eliminate its constants.

Variable Separable

Get all y with dy on one side, all x with dx on the other, then integrate.

Homogeneous

Substitute y = vx to make it separable, solve, then change back.

Linear

dy/dx + Py = Q is solved with IF = e^(∫ P dx).

Are You Exam-Ready?

8-Point Exam Quick-Check

State the order and degree of d²y/dx² + y = 0.

Solve dy/dx = 4x³.

Solve dy/dx = y by separation.

What does a particular solution require?

Find the integrating factor for dy/dx + 3y = x.

Form the differential equation of y = mx.

Solve dy/dx = x/y.

Solve dy/dx = cos x.

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Class 12 Mathematics Chapter 9: Differential Equations, Complete Notes and Practice

These free Class 12 Maths Differential Equations notes follow the NCERT 2026 to 27 syllabus and cover order and degree, general and particular solutions, forming differential equations, the variable separable method, homogeneous equations and linear equations with the integrating factor, with twelve worked examples and sixteen graded practice questions. Includes worked examples, step-by-step proofs and graded practice, free on SchoolRevise.com.