NCERT Solutions class 12 Maths Exercise 9.6 (Ex 9.6) Chapter 9 Differential Equations

NCERT Solutions for Class 12 Maths Exercise 9.6 Chapter 9 Differential Equations – FREE PDF Download

Free PDF download of NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.6 (Ex 9.6) and all chapter exercises at one place prepared by expert teacher as per NCERT (CBSE) books guidelines. Class 12 Maths Chapter 9 Differential Equations Exercise 9.6 Questions with Solutions to help you to revise complete Syllabus and Score More marks.

NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations (Ex 9.6) Exercise 9.6

In each of the following differential equations given in each Questions 1 to 4, find the general solution:

1.

Ans. Given: Differential equation

Comparing with , we have P = 2 and Q = sin x.

=> I.F. =

Solution is (I.F.) =

……….(i)

Applying product rule,

=> I =

Again applying product rule,

=> I =

I =

I =

I =

Putting the value of I in eq. (i),

=>

2.

Ans. Given: Differential equation

Comparing with ,

we have P = 2 and Q = .

=> I.F. =

Solution is

=> (I.F.) =

3.

Ans. Given: Differential equation

Comparing with ,

we have P = and Q = .

=> I.F. = = x

Solution is

=> (I.F.) =

4.

Ans. Given: Differential equation

Comparing with , we have P = and Q = .

I.F. =

Solution is

=> (I.F.) =

$=> y (s e c x + t a n x) = \int (s e c x . t a n x + s e c^{2} x - 1) d x + c$

For each of the following differential equations given in Question 5 to 8, find the general solution:

5.

Ans. Given: Differential equation

$=> \frac{d y}{d x} + \frac{y}{c o s^{2} x} = \frac{t a n x}{c o s^{2} x}$

Comparing with ,

we have P = and Q = .

=> I.F. =

Solution is

=> (I.F.) =

…….(i)

Putting and differentiating

Applying product rule,

Putting this value in eq. (i),

=>

divide by e^tanx, we get

y = (tanx -1) + ce^-tanx

6.

Ans. Given: Differential equation

Comparing with ,

we have P = and Q = .

=> I.F. =

Solution is

=> (I.F.) =

7.

Ans. Given: Differential equation

Comparing with ,

we have P = and Q = .

=> I.F. =

Solution is

=> (I.F.) =

Applying Product rule of Integration,

$=> y l o g x = 2 [(l o g x) \frac{x^{- 1}}{- 1} - \int \frac{1}{x} . \frac{x^{- 1}}{- 1} d x] + c$

8.

Ans. Given: Differential equation

Comparing with ,

we have P = and Q = .

=> I.F. =

Solution is

=> (I.F.) =

For each of the following differential equations given in Question 9 to 12, find the general solution:

9.

Ans. Given: Differential equation

Comparing with ,

we have P = and Q = .

=

=> I.F. =

Solution is

=> (I.F.) =

Applying product rule of Integration,

=

10.

Ans. Given: Differential equation

$\frac{d x}{d y} = x + y$

$\frac{d x}{d y} - x = y$

Comparing with , we have P = and Q = .

I.F. =

Solution is

=> x(I.F.) = $\int Q (I . F .) d y + c$

Applying product rule of Integration,

=> xe^-y=

=> xe^-y =

x+y+1 = ce^y

11.

Ans. Given: Differential equation

$\frac{d x}{d y} + \frac{1}{y} x = y$

Comparing with , w

e have P = and Q = .

I.F. =

Solution is x(I.F.) = $\int Q (I . F .) d y + c$

12.

Ans. Given: Differential equation

Comparing with ,

we have P = and Q = .

=> I.F. =

Solution is x(I.F.) = $\int Q (I . F .) d y + c$

For each of the differential equations given in Questions 13 to 15, find a particular solution satisfying the given condition:

13. when $x = \frac{π}{3}$

Ans. Given: Differential equation

when

Comparing with ,

we have P = and Q = .

=> I.F.=

Solution is (I.F.) =

………….. (1)

put value of x and y, we get

$=> \frac{0}{c o s^{2} \frac{π}{3}} = \frac{1}{c o s \frac{π}{3}} + c$

=> $0 = 2 + c$

=> c = -2 put value of c in (1), we get

$=> \frac{y}{c o s^{2} x} = \frac{1}{c o s x} - 2$

$=> y = c o s x - 2 c o s^{2} x$

14. when

Ans. Given: Differential equation

when

Comparing with ,

we have P = and Q = .

=> I.F.=

Solution is (I.F.) =

……….(i)

Now putting

Putting the value of in eq. (i),

=>

15. when

Ans. Given: Differential equation

Comparing with ,

we have P = and Q = .

I.F.=

Solution is (I.F.) =

……….(i)

Now putting in eq. (i),

Putting in eq. (i),

=>

=> y = 4sin³x – 2sin²x

16. Find the equation of the curve passing through the origin, given that the slope of the tangent to the curve at any point (x,y) is equal to the sum of coordinates of that point.

Ans. Slope of the tangent to the curve at any point (x,y) = Sum of coordinates of the point

Comparing with ,

we have P = and Q = .

=> I.F.=

Solution is

=> y(I.F.) =

Applying Product rule of Integration,

=>

……….(i)

Now, since curve (i) passes through the origin (0, 0),

therefore putting in eq. (i)

Putting in eq. (i),

17. Find the equation of the curve passing through the point (0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangents to the curve at that point by 5.

Ans. According to the question, Sum of the coordinates of any point say

on the curve

= Magnitude of the slope of the tangent to the curve + 5

Comparing with ,

we have P = and Q = .

=> I.F.=

Solution is

=> y(I.F.) =

Applying Product rule of Integration,

……….(i)

Now, since curve (i) passes through the point (0, 2),

therefore putting in eq. (i)

0+2 = 4+ce⁰

Putting in eq. (i),

18. Choose the correct answer:

The integrating factor of the differential equation is:

(A)

(B)

(C)

(D)

Ans. Given: Differential equation

Comparing with ,

we have P = and Q = .

=> I.F.=

Therefore, option (C) is correct.

19. Choose the correct answer:

The integrating factor of the differential equation $(1 - y^{2}) \frac{d x}{d y} + y x = a y (- 1 < y < 1) i s$

(A)

(B)

(C)

(D)

Ans. Given: Differential equation

(1 - y^{2}) \frac{d x}{d y} + y x = a y (- 1 < y < 1)

Comparing with ,

we have P = and Q =

=

=> I.F. =

Therefore, option (D) is correct.