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CBSE · Class 12 · Mathematics

Differential Equations

Standard difficulty · 20 questions · 40 marks

Name: Date: Score: ____ / 40

Show your working for Sections B and C. Answers are on the last page — try every question first!

Section A — Choose the correct answer8 × 1 = 8 marks
1.
The order and degree of the differential equation (y′)3 + 9y + 9 = 0 are:
  1. order 1, degree 3
  2. order 3, degree 1
  3. order 1, degree 4
  4. order 2, degree 3
2.
The general solution of dy/dx = 2x is:
  1. y = (2/3)x3 + C
  2. y = 2 + C
  3. y = x2 + C
  4. y = 2x2 + C
3.
The integrating factor of dy/dx − (1/x)y = x5 is:
  1. −log x
  2. x
  3. 1/x
  4. e−x
4.
The number of arbitrary constants in the general solution of a differential equation of order 3 is:
  1. 0
  2. 3
  3. 4
  4. 2
5.
The order of a differential equation is the order of:
  1. the lowest derivative in it
  2. the highest power of y
  3. the highest derivative in it
  4. the number of terms
6.
Which function is a solution of y″ + 16y = 0?
  1. y = x4
  2. y = e4x
  3. y = 5 sin 4x
  4. y = sin 16x
7.
The differential equation dy/dx = (8x2 + 7y2)/(2xy) is:
  1. linear in y
  2. variable separable
  3. homogeneous
  4. of order 2
8.
Assertion (A): y = Ce−9x is the general solution of dy/dx = 9y.
Reason (R): Separating the variables, dy/y = k dx gives log|y| = kx + c.
  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true, but R is false.
  4. A is false, but R is true.
Section B — Short answer8 × 2 = 16 marks
9.
Solve dy/dx = 7x2/(9y3).
10.
Solve dy/dx = 5y, given y = 8 when x = 0.
11.
Solve dy/dx + y = 4.
12.
Solve dy/dx + (1/x)y = x4, x > 0.
13.
Form the differential equation of the family y = A sin 3x + B cos 3x, where A and B are arbitrary constants.
14.
Solve the homogeneous equation x dy/dx = y − 7x.
15.
Solve dy/dx = 24x2 + 4, given y = −6 when x = 0.
16.
A population grows at a rate proportional to its size. It doubles in 16 years. In how many years will it become 16 times its present size?
Section C — Word problems4 × 4 = 16 marks
17.
Solve dy/dx + 4y = −9e2x, given y = 1 when x = 0.
18.
Solve x dy/dx − y = 5x2, given y = 7 when x = 1.
19.
Solve dy/dx = (x + 8)/(y − 7), given y = 0 when x = 0.
20.
A radioactive substance decays at a rate proportional to the amount present, and its half-life is 55 years. What fraction remains after 165 years? After how many years will only 1/8 of it remain?
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CBSE · Class 12 · Mathematics

Answer key

Differential Equations · AW-12M-DEQ-2026-V1 · For parents — fold back or tear off before handing the sheet to your child.

QAnswerWorking
1(a) order 1, degree 3highest derivative y′, raised to the power 3
2(c) y = x2 + Cintegrate both sides
3(c) 1/xe^(∫P dx) with P = −1/x
4(b) 3one constant for each order
5(c) the highest derivative in it—
6(c) y = 5 sin 4x(sin 4x)″ = −16 sin 4x
7(c) homogeneousevery term has degree 2, so dy/dx is a function of y/x
8(d) A is false, but R is true.A is false: the solution is y = Ce9x, not Ce−9x. R is true.
9(9/4)y4 = (7/3)x3 + C9y3 dy = 7x2 dx
10y = 8e5xy = Ce5x; x = 0 gives C = 8
11y = 4 + Ce−xintegrating factor ex; yex = 4ex + C
12y = (1/6)x5 + C/xintegrating factor x; d/dx(xy) = x5
13y″ + 9y = 0differentiate twice and eliminate the constants
14y = −7x log|x| + Cxput y = vx: x dv/dx = −7, so v = −7 log|x| + C
15y = 8x3 + 4x − 6y = 8x3 + 4x + C; C = −6
1664 yearsP = P0e^(kt) with e^(16k) = 2, so 16 = 24 needs t = 4 × 16
17y = −(3/2)e2x + (5/2)e−4xintegrating factor e4x: ye4x = −(3/2)e6x + C; C = 1 − (−3/2) = 5/2
18y = 5x2 + 2xdivide by x2: d/dx(y/x) = 5, so y/x = 5x + C; C = 2
19y2 − 14y = x2 + 16x(y − 7) dy = (x + 8) dx ⇒ y2/2 − 7y = x2/2 + 8x + C; C = 0
201/8 remains after 165 years; 1/8 remains after 165 yearsA = A0e^(−λt) with e^(−55λ) = 1/2, so A = A0(1/2)^(t/55)