# 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

**Answer.** The given differential equation is

Order = 2 and Degree = 1

∴ Order + Degree = 2 + 1 = 3

** Question. Write the sum of the order an**d

**degree of the differential equation**

**Answer.** Order = 2, Degree = 2.

∴ Order + Degree = 2 + 2 = 4

** Question.** Write the degree of the differential equation

**Answer.** Degree of given differential equation is 2.

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

**Answer.** Degree of the given differential equation is 1.

**Question.** Find the differential equation representing the**family of curves**

**where A and B are arbitrary constants.****Answer.**

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

**Answer.** The given differential equation is

**Question.** Write the solution of the differential equation

**Answer.**

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

**Answer.**

**Question.** Solve the following differential equation :**x cos y dy = (xe ^{x} log x + ex) dx.**

**Answer.**We have, x cos y dy = (xe

^{x}log x + e

^{x}) dx

**Question.** Solve the following differential equation :**tan y dx + sec ^{2} y tan x dy = 0.**

**Answer.**We have, sec

^{2}y tan x dy = –tan y dx

⇒ 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 :****sec ^{2} x tan y dx + sec^{2} y tan x dy = 0.**

**Answer.**We have, sec

^{2}x tan y dx + sec

^{2}y tan x dy = 0

sec

^{2}y tan x dy = – sec

^{2}x tan y dx

**Question.** Solve the following differential equation :

**Answer.**

Integrating both sides, we get

log(1 + y^{2}) = –log(1 – x^{2}) + log C

⇒ log(1 + y^{2}) + log(1 – x^{2}) = log C

⇒ log(1 – x^{2})(1 + y^{2}) = log C

⇒ (1 – x^{2})(1 + y^{2}) = C

**Short Answer Type Questions**

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

**Answer.** We have, y = 3cos(logx) + 4sin(logx) …(i)

Differentiating (i) w.r.t. x, we get

**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 II^{nd} quadrant touching co-ordinate axes is

(x + a)^{2} + (y – a)^{2} = a^{2} …(i)

[Here C is (–a, a) and radius = a]

which has only one arbitrary constant a.

Differentiating (i) w.r.t. x, we get

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

Equation of parabola having vertex at origin and axis along positive y-axis is

x^{2} = 4ay, where a is the parameter. …(i)

Differentiating (i) w.r.t. x, we get

**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

Therefore, equation of family C is (x – a)^{2} + y^{2} = a^{2}

or x^{2} + y^{2} = 2ax …(i)

where, a is any arbitrary constant.

Differentiating (i) w.r.t. x, we get

which is the required differential equation.

**Question.** Solve the differential equation

**Answer.**

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

**Answer.**

**Question.** Find the general solution of the differential**equation**

**Answer.**

**Question.** Find the particular solution of the differential

**Answer.**

**Question.** Solve the differential equation

**Answer.**

**Question.** Solve the following differential equation :

**Answer.**

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

**Answer.**

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 :**

**Answer.**

**Question.** Find the particular solution of the following differential equation; dy/dx =1+x^{2}+y^{2}+x^{2}y^{2}, given that y =1 when x = 0.**Answer.**

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

**Answer.**

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

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

**Answer.**

**Question.****Find the particular solution of the di fferential equation :**

**dy/dx + y cot x = 2x +x**

^{2}cot x, x ≠ 0, given that y = 0,**when x = π/2.**

**Answer.**

**Question.** Solve the following di**ff**erential equation

**Answer.**

**Question.** Solve the following differential equation :**Answer.**

**Question.** Solve the following differential equation :**Answer.**

**Question.****Solve the following differential equation :****(1 + y ^{2}) (1 + log x) dx + xdy = 0. **

**Answer.**We have, (1 + y

^{2})(1 + logx)dx + xdy = 0

⇒ (1 + y

^{2})(1 + logx)dx = –xdy

**Question.** Solve the following differential equation

**Answer.**

**Question.**

**Answer.**

**Question.** Solve the following differential equation :

(x^{2} – y^{2})dx + 2xydy = 0, given that y = 1, when x = 1.**Answer.** We have, (x^{2} – y^{2})dx + 2xy dy = 0

**Question.** Solve the following differential equation :

**Answer.**

**Question.****Solve the following differential equation :**

**Answer.**

**Question.** Solve the following differential equation :**(1 + e ^{2x})dy + (1 + y^{2})e^{x}dx = 0.**

**Answer.**We have, (1 + e

^{2x})dy + (1 + y

^{2})e

^{x}dx = 0

**Question.** Solve the differential equation :**dy/dx + 2y = 6e ^{x} .**

**Answer.**

**Question.** Solve the following differential equation :**x cos ydy = (xe ^{x} log x + e^{x})dx.**

**Answer.**We have, xcosydy = (xe

^{x}logx + e

^{x})dx

**Question.** Solve the following differential equation :

**Answer.**

**Long Answer Type Questions**

**Question.** Solve the differential equation

**Answer.**

**Question.** Show that the differential equation

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

**Question.** Show that the differential equation 2ye^{x/y} dx + (y – 2x e^{x/y}) dy = 0 is homogeneous.

Find the particular solution of this differential equation, given that x = 0 when y = 1**Answer.** We have, 2y e^{x}/ y dx + (y − 2x e^{x}/ y )dy = 0

**Question**. Show that the differential equation (x e^{x}/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 e^{x}/y + y) dx = x dy

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

**Question.****Find the particular solution of the differential equation (3xy + y ^{2}) dx + (x^{2} + xy) dy = 0 : for x = 1, y = 1**

**Answer.**We have, (3xy + y

^{2}) dx + (x

^{2}+ xy) dy = 0

⇒ 2x^{2}y^{2} + 4x^{3}y = C [where C = (C’)4] …(ii)

Put x = 1, y = 1 in (ii), we get ⇒C = 6

Hence 2x^{2}y^{2} + 4x^{3}y = 6 ⇒ x^{2}y^{2} + 2x^{3}y = 3

is the required particular solution.

**Question.** Find the particular solution of the following differential equation given that y = 0 when x = 1 : (x^{2} + xy) dy = (**x ^{2}** +

**y**) dx

^{2}**Answer.**We have, (x

^{2}+ xy) dy = (x

^{2}+ y

^{2}) dx

**Question.** Solve the following differential equation :

**Answer.** We have,