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Differential Equations – Class 12 Mathematics notes, important questions and MCQs

NCERT Class 12 Mathematics, chapter 9. Notes, key terms, important questions with answers and MCQs.

Chapter overview

A differential equation relates a function with its derivatives. Its order and degree classify it, and methods like separation of variables and integrating factors solve it.

Basics

Order: highest derivative present; degree: power of the highest derivative (when the equation is a polynomial in derivatives).
General solution has as many arbitrary constants as the order; particular solution uses given conditions.

Methods

Variables separable: f(y) dy = g(x) dx, integrate both sides.
Homogeneous equations: put y = vx.
Linear: dy/dx + Py = Q, integrating factor IF = e^(∫P dx), solution y·IF = ∫Q·IF dx + C.

Key terms

Order: order of the highest derivative in the equation
Integrating factor: e^(∫P dx), multiplier that makes a linear equation exact
Particular solution: solution obtained from the general solution by using given initial conditions to fix the constants

Formulas

IF = e^(∫P dx)
y·IF = ∫Q·IF dx + C

Important questions with answers

Form the differential equation of the family of lines y = mx, where m is arbitrary.2 marks

dy/dx = m = y/x, so x dy/dx − y = 0.

Solve the homogeneous equation (x² + y²) dx = 2xy dy.3 marks

Put y = vx: (1 + v²) = 2v(v + x dv/dx), so x dv/dx = (1 − v²)/(2v). Then 2v dv/(1 − v²) = dx/x, giving −log|1 − v²| = log|x| + log C, i.e. x(1 − v²) = C₁. Replacing v = y/x: x² − y² = Cx.

Solve dy/dx + y cot x = 2x + x² cot x, given y(π/2) = 0.5 marks

Linear with P = cot x, IF = e^(∫cot x dx) = sin x. y sin x = ∫(2x + x² cot x) sin x dx = ∫(2x sin x + x² cos x) dx = x² sin x + C (since d/dx(x² sin x) = 2x sin x + x² cos x). So y = x² + C cosec x. y(π/2) = 0: π²/4 + C = 0, C = −π²/4. Solution: y = x² − (π²/4) cosec x.

Quick revision

Number of constants = order.
Linear equation needs IF.
Homogeneous: y = vx.

Exam tips

Degree is not defined if the derivative appears inside sin, log, etc.
Identify the type first: variables separable, homogeneous (put y = vx) or linear (find IF).

Practice MCQs

Order of d²y/dx² + 3(dy/dx)³ = 0 is
1. 1
2. 2
3. 3
4. 6
Show answer
Answer: (b) 2. Highest derivative is second.
Degree of (d²y/dx²)³ + dy/dx = 0 is
1. 1
2. 3
3. 2
4. not defined
Show answer
Answer: (b) 3. Power of highest derivative.
Solution of dy/dx = y is
1. y = e^(−x) + C
2. y = x + C
3. y = Cx
4. y = Ceˣ
Show answer
Answer: (d) y = Ceˣ. dy/y = dx.
Integrating factor of dy/dx + y/x = x² is
1. 1/x
2. x
3. eˣ
4. log x
Show answer
Answer: (b) x. e^(log x).
The general solution of dy/dx = e^(x+y) is
1. e^x − e^y = C
2. e^x + e^(−y) = C
3. e^(x+y) = C
4. e^(−x) + e^y = C
Show answer
Answer: (b) e^x + e^(−y) = C. e^(−y) dy = e^x dx.
Number of arbitrary constants in the general solution of a 3rd-order differential equation is
1. 1
2. 3
3. 2
4. 0
Show answer
Answer: (b) 3. Equals order.
Solution of dy/dx = x/y with y(0) = 2 is
1. y = x + 2
2. y² + x² = 4
3. y² − x² = 4
4. xy = 2
Show answer
Answer: (c) y² − x² = 4. y dy = x dx ⇒ y² − x² = C = 4.
The order and degree of (dy/dx)² + y = sin(d²y/dx²) is
1. 2 and 1
2. 2 and not defined
3. 1 and 2
4. 2 and 2
Show answer
Answer: (b) 2 and not defined. Derivative inside sine.

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Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.