Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Please refer to below Exam Question for Class 12 Mathematics Chapter 9 Differential Equations. These questions and answers have been prepared by expert Class 12 Mathematics teachers based on the latest NCERT Book for Class 12 Mathematics and examination guidelines issued by CBSE, NCERT, and KVS. We have provided Class 12 Mathematics exam questions for all chapters in your textbooks. You will be able to easily learn problems and solutions which are expected to come in the upcoming class tests and exams for standard 10th.

Chapter 9 Differential Equations Class 12 Mathematics Exam Question

All questions and answers provided below for Exam Question Class 12 Mathematics Chapter 9 Differential Equations are very important and should be revised daily.

Exam Question Class 12 Mathematics Chapter 9 Differential Equations

Very Short Answer Type Questions

Question. Write the sum of the order and degree of the following differential equation

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer. The given differential equation is

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Order = 2 and Degree = 1
∴ Order + Degree = 2 + 1 = 3

Question. Write the sum of the order and degree of the differential equation

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer. Order = 2, Degree = 2.
∴ Order + Degree = 2 + 2 = 4

Question. Write the degree of the differential equation

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer. Degree of given differential equation is 2.

Question. What is the degree of the following differential equation?

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer. Degree of the given differential equation is 1.

Question. Find the differential equation representing the family of curves

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

where A and B are arbitrary constants.
Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Write the integrating factor of the following differential equation :

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer. The given differential equation is

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Write the solution of the differential equation

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Taking log on both sides to the base 2, we get
⇒ log2 2y = log2 [(C + x) log2]
⇒ y = log2 [(C + x) log2]
which is the required solution

Question. Find the solution of the differential equation

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the following differential equation :
x cos y dy = (xex log x + ex) dx.
Answer. We have, x cos y dy = (xex log x + ex) dx

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the following differential equation :
tan y dx + sec2 y tan x dy = 0.
Answer. We have, sec2 y tan x dy = –tan y dx

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

⇒ log |t| = − log |sin x| + log C
⇒  log |tan y| + log |sin x| = log C
⇒  log |tan y · sin x| = log C
⇒  tan y sin x = C

Question. Solve the following differential equation :
sec2 x tan y dx + sec2 y tan x dy = 0.
Answer. We have, sec2 x tan y dx + sec2 y tan x dy = 0
sec2 y tan x dy = – sec2 x tan y dx

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the following differential equation :

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Integrating both sides, we get
log(1 + y2) = –log(1 – x2) + log C
⇒ log(1 + y2) + log(1 – x2) = log C
⇒ log(1 – x2)(1 + y2) = log C
⇒ (1 – x2)(1 + y2) = C

Short Answer Type Questions

Question. Verify that y = 3cos(logx) + 4sin(logx) is a solution of the differential equation

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer. We have, y = 3cos(logx) + 4sin(logx)                      …(i)
Differentiating (i) w.r.t. x, we get

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
Answer. The equation of the circles in IInd quadrant touching co-ordinate axes is

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

(x + a)2 + (y – a)2 = a2                      …(i)
[Here C is (–a, a) and radius = a]
which has only one arbitrary constant a.
Differentiating (i) w.r.t. x, we get

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations
Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Equation of parabola having vertex at origin and axis along positive y-axis is
x2 = 4ay, where a is the parameter.                      …(i)
Differentiating (i) w.r.t. x, we get

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Find the differential equation of the family of all circles touching the y-axis at the origin.
Answer. Let C denote the family of circles touching y-axis at the origin. Let (a, 0) be the co-ordinates of the centre of any member of the family

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Therefore, equation of family C is (x – a)2 + y2 = a2
or x2 + y2 = 2ax                      …(i)
where, a is any arbitrary constant.
Differentiating (i) w.r.t. x, we get

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

which is the required differential equation.

Question. Solve the differential equation

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations
Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. If y(x) is a solution of the differential equation

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Find the general solution of the differential equation

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations
Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Find the particular solution of the differential

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the differential equation

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations
Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the following differential equation :

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Find the particular solution of the following differential equation :

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Integrating both sides, we get
y – 2log(y + 2) = x + 2log x + C
when x = 1, y = – 1
So, – 1 – 2log (– 1 + 2) = 1 + 2 log 1 + C
⇒ C = – 1 – 1 = – 2
So, we have y – 2log(y + 2) = x + 2log x – 2
⇒ y – x + 2 = 2log x(y + 2).

Question. Solve the following differential equation :

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Find the particular solution of the following differential equation; dy/dx =1+x2+y2+x2y2, given that y =1 when x = 0.
Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Find the particular solution of the following differential equation :

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the following differential equation :
dy/dx − y = cos x, given that if x = 0, y = 1.
Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Find the particular solution of the following differential equation, given that x = 2, y = 1 :
x (dy/dx) + 2y = x2 ,(x ≠ 0)
Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Find the particular solution of the differential equation :
dy/dx + y cot x = 2x +x2 cot x, x ≠ 0, given that y = 0, when x = π/2. 
Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the following differential equation

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the following differential equation :
Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations
Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the following differential equation :
Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the following differential equation :
(1 + y2) (1 + log x) dx + xdy = 0. 
Answer. We have, (1 + y2)(1 + logx)dx + xdy = 0
⇒ (1 + y2)(1 + logx)dx = –xdy

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the following differential equation

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the following differential equation :
(x2 – y2)dx + 2xydy = 0, given that y = 1, when x = 1.

Answer. We have, (x2 – y2)dx + 2xy dy = 0

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the following differential equation :

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations
Exam Question for Class 12 Mathematics Chapter 9 Differential Equations
Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the following differential equation :

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the following differential equation :
(1 + e2x)dy + (1 + y2)exdx = 0.
Answer. We have, (1 + e2x)dy + (1 + y2)ex dx = 0

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the differential equation :
dy/dx + 2y = 6ex .
Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the following differential equation :
x cos ydy = (xex log x + ex)dx.
Answer. We have, xcosydy = (xex logx + ex)dx

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the following differential equation :

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations
Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Long Answer Type Questions

Question. Solve the differential equation

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Show that the differential equation

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

is homogeneous. Find the particular solution of this differential equation, given that x = 1 when y = π/2
Answer.

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations
Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Show that the differential equation 2yex/y dx + (y – 2x ex/y) dy = 0 is homogeneous.
Find the particular solution of this differential equation, given that x = 0 when y = 1

Answer. We have, 2y ex/ y dx + (y − 2x ex/ y )dy = 0

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Show that the differential equation (x ex/y+ y) dx = xdy is homogeneous. Find the particular solution of this differential equation, given that x = 1 when y = 1
Answer. We have, (x ex/y + y) dx = x dy

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations
Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Find the particular solution of the differential equation dx/dy + x cot y = 2y + y2 cot y,(y ≠ 0), given that x = 0 when y = π/2.
Answer. We have,

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Find the particular solution of the differential equation (3xy + y2) dx + (x2 + xy) dy = 0 : for x = 1, y = 1
Answer. We have, (3xy + y2) dx + (x2 + xy) dy = 0

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations
Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

⇒ 2x2y2 + 4x3y = C [where C = (C’)4]                        …(ii)
Put x = 1, y = 1 in (ii), we get ⇒C = 6
Hence 2x2y2 + 4x3y = 6 ⇒ x2y2 + 2x3y = 3
is the required particular solution.

Question. Find the particular solution of the following differential equation given that y = 0 when x = 1 : (x2 + xy) dy = (x2 + y2) dx
Answer. We have, (x2 + xy) dy = (x2 + y2) dx

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Question. Solve the following differential equation :

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

Answer. We have,

Exam Question for Class 12 Mathematics Chapter 9 Differential Equations

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