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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image.png)
Answer. The given differential equation is
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-15.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-1.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-7.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-8.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-9.png)
where A and B are arbitrary constants.
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-16.png)
Question. Write the integrating factor of the following differential equation :
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-11.png)
Answer. The given differential equation is
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-20.png)
Question. Write the solution of the differential equation
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-12.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-21.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-13.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-22.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-23.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-24.png)
⇒ 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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-25.png)
Question. Solve the following differential equation :
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-14.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-26.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-27.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-57.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-58.png)
(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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-59.png)
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-60.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-61.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-62.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-63.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-64.png)
which is the required differential equation.
Question. Solve the differential equation
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-32.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-87.png)
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-88.png)
Question. If y(x) is a solution of the differential equation
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-33.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-89.png)
Question. Find the general solution of the differential equation
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-34.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-90.png)
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-91.png)
Question. Find the particular solution of the differential
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-35.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-92.png)
Question. Solve the differential equation
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-38.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-93.png)
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-94.png)
Question. Solve the following differential equation :
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-37.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-95.png)
Question. Find the particular solution of the following differential equation :
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-36.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-96.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-39.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-97.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-98.png)
Question. Find the particular solution of the following differential equation :
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-42.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-104.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-105.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-106.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-107.png)
Question. Solve the following differential equation
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-43.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-108.png)
Question. Solve the following differential equation :
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-109.png)
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-110.png)
Question. Solve the following differential equation :
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-111.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-112.png)
Question. Solve the following differential equation
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-52.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-130.png)
Question.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-53.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-131.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-132.png)
Question. Solve the following differential equation :
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-54.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-133.png)
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-134.png)
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-135.png)
Question. Solve the following differential equation :
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-55.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-136.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-137.png)
Question. Solve the differential equation :
dy/dx + 2y = 6ex .
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-138.png)
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](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-139.png)
Question. Solve the following differential equation :
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-56.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-140.png)
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://dkgoelsolutions.com/wp-content/uploads/2022/06/image-141.png)
Long Answer Type Questions
Question. Solve the differential equation
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://assignmentsbag.com/wp-content/uploads/2022/06/image-74.png)
Answer.
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://assignmentsbag.com/wp-content/uploads/2022/06/image-95.png)
Question. Show that the differential equation
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://assignmentsbag.com/wp-content/uploads/2022/06/image-73.png)
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](https://assignmentsbag.com/wp-content/uploads/2022/06/image-96.png)
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://assignmentsbag.com/wp-content/uploads/2022/06/image-97.png)
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](https://assignmentsbag.com/wp-content/uploads/2022/06/image-100.png)
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](https://assignmentsbag.com/wp-content/uploads/2022/06/image-101.png)
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://assignmentsbag.com/wp-content/uploads/2022/06/image-102.png)
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](https://assignmentsbag.com/wp-content/uploads/2022/06/image-103.png)
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](https://assignmentsbag.com/wp-content/uploads/2022/06/image-83.png)
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://assignmentsbag.com/wp-content/uploads/2022/06/image-84.png)
⇒ 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](https://assignmentsbag.com/wp-content/uploads/2022/06/image-82.png)
Question. Solve the following differential equation :
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://assignmentsbag.com/wp-content/uploads/2022/06/image-105.png)
Answer. We have,
![Exam Question for Class 12 Mathematics Chapter 9 Differential Equations](https://assignmentsbag.com/wp-content/uploads/2022/06/image-104.png)