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**NCERT Solutions for Class 12 Maths Differential Equations Exercise 9.6**## NCERT Solutions for Class 12 Maths Differential Equations

For each of the differential equations given in Exercises 1 to 12, find the general solution:

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 1.**

**NCERT Solutions:**

Given differential equation is .

This equation is a type of , where P = 2 and Q = sin x.

Now, the Integrating Factor (IF) is .

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

…(1)

Putting the value of I in equation (1), we get

This is the required general solution of the given differential equation.

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 2.**

**NCERT Solutions:**

Given differential equation is .

This equation is a type of , where P = 3 and Q = e^{-2x}.

Now, the Integrating Factor (IF) is .

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

…(1)

Now,

This is the required general solution of the given differential equation.

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 3.**

**NCERT Solutions:**

Given differential equation is .

This equation is a type of , where P = 1/x and Q = x^{2}.

Now, the Integrating Factor (IF) is .

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

…(1)

Now,

This is the required general solution of the given differential equation.

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 4.**

**NCERT Solutions:**

Given differential equation is .

This equation is a type of , where P = sec x and Q = tan x.

Now, the Integrating Factor (IF) is .

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

…(1)

Now,

This is the required general solution of the given differential equation.

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 5.**

**NCERT Solutions:**

Given differential equation is .

This equation is a type of , where P = sec^{2} x and Q = tan x sec^{2} x.

Now, the Integrating Factor (IF) is.

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

…(1)

Putting the value of I in equation (1), we get

This is the required general solution of the given differential equation.

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 6.**

**NCERT Solutions:**

Given differential equation is.

This equation is a type of , where P = 2/x and Q = x log x .

Now, the Integrating Factor (IF) is .

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

…(1)

This is the required general solution of the given differential equation.

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 7.**

**NCERT Solutions:**

Given differential equation is .

This equation is a type of , where P = 1/(x log x) and Q = 2/x^{2} .

Now, the Integrating Factor (IF) is.

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

…(1)

This is the required general solution of the given differential equation.

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 8.**

(1 + x^{2}) dy + 2xy dx = cot x dx (x ≠ 0)

**NCERT Solutions:**

Given differential equation is (1 + x^{2}) dy + 2xy dx = cot x dx (x ≠ 0).

This equation is a type of , where P = 2x/(1 + x^{2}) and Q = (cot x)/(1 + x^{2}) .

Since

Then, the Integrating Factor (IF) is .

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

…(1)

This is the required general solution of the given differential equation.

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 9.**

**NCERT Solutions:**

Given differential equation is .

This equation is a type of , where P = 1/x + cot x and Q = 1.

Then, the Integrating Factor (IF) is .

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

…(1)

This is the required general solution of the given differential equation.

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 10.**

**NCERT Solutions:**

Given differential equation is . Or,

This equation is a type of , where P = -1 and Q = y.

Then, the Integrating Factor (IF) is .

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

…(1)

This is the required general solution of the given differential equation.

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 11.**

y dx + (x – y^{2}) dy = 0

**NCERT Solutions:**

Given differential equation is y dx + (x – y^{2}) dy = 0

This equation is a type of , where P = 1/y and Q = y.

Then, the Integrating Factor (IF) is .

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

…(1)

This is the required general solution of the given differential equation.

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 12.**

**NCERT Solutions:**

Given differential equation is

This equation is a type of , where P = -1/y and Q = 3y.

Then, the Integrating Factor (IF) is .

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

…(1)

This is the required general solution of the given differential equation.

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

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 13.**

**NCERT Solutions:**

Given differential equation is .

This equation is a type of , where P = 2tan x and Q = sin x.

Since,

Then, the Integrating Factor (IF) is

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

…(1)

Since y = 0 at x = π/3, then

This is the required particular solution of the given differential equation.

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 14.**

**NCERT Solutions:**

Given differential equation is .

This equation is a type of , where .

Since,

Then, the Integrating Factor (IF) is

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

…(1)

Since y = 0 at x = 1, then

This is the required particular solution of the given differential equation.

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 15.**

**NCERT Solutions:**

Given differential equation is .

This equation is a type of , where P = -3cot x and Q = sin 2x.

Since,

Then, the Integrating Factor (IF) is

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

Since y = 2 at x = π/2, then

Hence,

This is the required particular solution of the given differential equation.

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 16.**

Find the equation of a 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 the coordinates of the point.

**NCERT Solutions:**

Since the slope of a tangent to a curve is given by dy/dx.

Now, from the given information, we have

This equation is a type of , where P = -1 and Q = x.

Now, the Integrating Factor (IF) is

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

It is given that the curve passes through the origin (0, 0). We have x = 0, y = 0.

Hence,

This is the required particular solution of the given curve.

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 17.**

Find the equation of a 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 tangent to the curve at that point by 5.

**NCERT Solutions:**

Since the slope of a tangent to a curve is given by dy/dx.

Now, from the given information, we have

This equation is a type of , where P = -1 and Q = x – 5.

Now, the Integrating Factor (IF) is

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

It is given that the curve passes through the origin (0, 2).

We have x = 0, y = 2. 2 = -0 + Ce^{0}, then C = -2.

Hence,

This is the required particular solution of the given curve.

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 18.**

The Integrating Factor of the differential equation is

(A) e^{–x }

(B) e^{–y }

(C) 1/x

(D) x

**NCERT Solutions:**

Given differential equation is or, .

This equation is a type of , where P = -1/x and Q = 2x.

Then, the Integrating Factor (IF) is

**NCERT Solutions for Class 12 Maths Differential Equations ****Exercise**** 9.6: Ques No 19.**

The Integrating Factor of the differential equation is

**NCERT Solutions:**

Given differential equation can be written as

This equation is a type of , where .

Then, the Integrating Factor (IF) is