# NCERT Solutions for Class 12 Maths Differential Equations Miscellaneous Exercise

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## NCERT Solutions for Class 12 Maths Differential Equations Miscellaneous Exercise

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 1**

For each of the differential equations given below, indicate its order and degree (if defined).

(i)

**NCERT Solutions:**

Given differential equation is .

The highest order derivative present in the given differential equation is , so its order is **TWO**.

The given differential equation is a polynomial equation in its derivatives and the highest power of the derivative is one. So, its degree is **ONE**.

(ii)

**NCERT Solutions:**

Given differential equation is .

The highest order derivative present in the given differential equation is , so its order is **ONE**.

The given differential equation is a polynomial equation in its derivatives and the highest power of the derivative is three. So, its degree is **THREE**.

(iii)

**NCERT Solutions:**

Given differential equation is .

The highest order derivative present in the given differential equation is , so its order is **FOUR**.

The given differential equation is not a polynomial equation in its derivatives and so its degree is **NOT Defined**.

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 2**

For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

(i) Check whether the function y = ae^{x} + be^{–x} + x^{2} is the solution of .

**NCERT Solutions:**

Given function is y = ae^{x} + be^{–x} + x^{2} …(1)

Differentiating the equation (1) with respect to x, we get

…(2)

Differentiating the equation (2) with respect to x, we get

…(3)

Now,

Thus, the given function y = ae^{x} + be^{–x} + x^{2} is **NOT** the solution of .

(ii) Check whether the function y = e^{x} (a cos x + b sin x) is the solution of .

**NCERT Solutions:**

Given function is y = e^{x}(a cos x + b sin x) …(1)

Differentiating the equation (1) with respect to x, we get

…(2)

Differentiating the equation (2) with respect to x, we get

…(3)

Now,

Thus, the given function y = e^{x} (a cos x + b sin x) is the solution of .

(iii) Check whether the function y = x sin 3x is the solution of .

**NCERT Solutions:**

Given function is y = x sin 3x …(1)

Differentiating the equation (1) with respect to x, we get

…(2)

Differentiating the equation (2) with respect to x, we get

…(3)

Now,

Thus, the given function y = x sin 3x is the solution of .

(iv) Check whether the function x^{2} = 2y^{2} log y is the solution of .

**NCERT Solutions:**

Given function is x^{2} = 2y^{2} log y …(1)

Differentiating the equation (1) with respect to x, we get

…(2)

From (1),

Thus, the given function x^{2} = 2y^{2} log y is the solution of .

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 3**

Form the differential equation representing the family of curves given by (x – a)^{2} + 2y^{2} = a^{2}, where a is an arbitrary constant.

**NCERT Solutions:**

Given equation is (x – a)^{2} + 2y^{2} = a^{2} …(1)

Differentiating (1) with respect to x, we get

Putting in (1), we get

Thus, the differential equation of the given equation is .

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 4**

Prove that x^{2} – y^{2} = c (x^{2} + y^{2})^{2} is the general solution of differential equation (x^{3} – 3xy^{2}) dx = (y^{3} – 3x^{2}y) dy, where c is a parameter.

**NCERT Solutions:**

Given differential equation is

…(1)

In equation (1), we see that dy/dx is in form of g(y/x), so it is a Homogeneous function of degree zero. Thus, the given differential equation is Homogeneous Differential Equation.

To solve this, we need to put y = vx …(2)

Differentiating the equation (2) with respect to x, we get

On substituting y = vx and in equation (1), we get

On squaring both sides, we get x^{2} – y^{2} = c(x^{2} + y^{2})^{2 }where c = C’^{2} is an arbitrary constant.

Thus, x^{2} – y^{2} = c(x^{2} + y^{2})^{2} is the general solution of the given differential equation.

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 5**

Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

**NCERT Solutions:**

Let the equation of the family of the circles which touch the co-ordinates axes in the first quadrant is

(x – a)^{2} + (y – a)^{2} = a^{2} …(1)

, where a is the radius of the circle.

Differentiating (1) with respect to x, we get

Substituting the value of ‘a’ in (1), we get

This is required differential equation of the family of circles in the first quadrant which touch the coordinate axes.

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 6**

Find the general solution of the differential equation .

**NCERT Solutions:**

Given differential equation is

.

Integrating both sides, we get

This is required general solution of the given differential equation.

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 7**

Show that the general solution of the differential equation is given by (x + y + 1) = A (1 – x – y – 2xy), where A is parameter

**NCERT Solutions:**

Given differential equation is .

Integrating both sides, we get

Thus, the general solution of the given differential equation is (x + y + 1) = A (1 – x – y – 2xy).

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 8**

Find the equation of the curve passing through the point whose differential equation is sin x cos y dx + cos x sin y dy = 0.

**NCERT Solutions:**

Given differential equation is sin x cos y dx + cos x sin y dy = 0.

Integrating both sides, we get

⇒ –log |cos x| – log|cos y| = –log|C|

⇒ –log |cos x cos y| = –log|C|

⇒ cos x cos y = C

Since the curve passes through .

Thus,

This is required equation of the given curve.

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 9**

Find the particular solution of the differential equation (1 + e^{2x}) dy + (1 + y^{2}) e^{x} dx = 0, given that y = 1 when x = 0.

**NCERT Solutions:**

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 10**

Solve the differential equation .

**NCERT Solutions:**

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 11**

Find a particular solution of the differential equation (x – y) (dx + dy) = dx – dy, given that y = –1, when x = 0. (Hint: put x – y = t)

**NCERT Solutions:**

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 12**

Solve the differential equation .

**NCERT Solutions:**

Given differential equation is …(1)

This equation is a type of , where .

Now, the Integrating Factor (IF) is .

Thus, the solution of the given differential equation is .

Putting the values of IF and Q, we get

This is required general solution of the given differential equation.

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 13**

Find a particular solution of the differential equation (x ≠ 0), given that y = 0 when x = π/2

**NCERT Solutions:**

Given differential equation is …(1)

This equation is a type of , where P = cot x and Q = 4x cosec 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

This is required particular solution of the given differential equation.

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 14**

Find a particular solution of the differential equation (x + 1)(dy/dx) = 2 e^{–y} – 1, given that y = 0 when x = 0.

**NCERT Solutions:**

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 15**

The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20, 000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?

**NCERT Solutions:**

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 16**

The general solution of the differential equation is

(A) xy = C

(B) x = Cy^{2 }

(C) y = Cx

(D) y = Cx^{2}

**NCERT Solutions:**

Given differential equation is .

Integrating both sides, we get

This is required general solution.

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 17**

The general solution of a differential equation of the type is

**NCERT Solutions:**

**NCERT Solutions Class 12 Math Differential Equations Miscellaneous ****Exercise**** Ques No 18**

The general solution of the differential equation e^{x} dy + (ye^{x} + 2x) dx = 0 is

(A) x e^{y} + x^{2} = C

(B) x e^{y} + y^{2} = C

(C) y e^{x} + x^{2} = C

(D) y e^{y} + x^{2} = C

**NCERT Solutions:**