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## Differential Equations Class 12 MCQs Questions with Answers

Don’t forget to practice the multitude of MCQ Questions on Differential Equations Class 12 with answers so you can apply your skills during the exam.

Question 1.

The degree of the differential equation:

(\(\frac { d^2y }{dx^2}\))³ + (\(\frac { dy }{dx}\))² + sin (\(\frac { dy }{dx}\)) + 1 = 0 is

(a) 3

(b) 2

(c) 1

(d) not defined.

## Answer

Answer: (a) 3

Question 2.

The order of the differential equation:

2x² \(\frac { d^2y }{dx^2}\) – 3 \(\frac { dy }{dx}\) + y = 0 is

(a) 2

(b) 1

(c) 0

(d) not defined.

## Answer

Answer: (a) 2

Question 3.

The number of arbitrary constants in the general solution of a differential equation of fourth order is:

(a) 0

(b) 2

(c) 3

(d) 4.

## Answer

Answer: (d) 4.

Question 4.

The number of arbitrary constants in the particular solution of a differential equation of third order is:

(a) 3

(b) 2

(c) 1

(d) 0.

## Answer

Answer: (d) 0.

Question 5.

Which of the following differential equations has y = c_{1} e^{x}+ c_{2} e^{-x} as the general solution?

(a) \(\frac { d^2y }{dx^2}\) + y = 0

(b) \(\frac { d^2y }{dx^2}\) – y = 0

(c) \(\frac { d^2y }{dx^2}\) + 1 = 0

(d) \(\frac { d^2y }{dx^2}\) – 1 = 0

## Answer

Answer: (b) \(\frac { d^2y }{dx^2}\) – y = 0

Question 6.

Which of the following differential equations has y = x as one of its particular solutions?

(a) \(\frac { d^2y }{dx^2}\) – x² \(\frac { dy }{dx}\) + xy = x

(b) \(\frac { d^2y }{dx^2}\) + x \(\frac { dy }{dx}\) + xy = x

(c) \(\frac { d^2y }{dx^2}\) – x² \(\frac { dy }{dx}\) + xy = 0

(d) \(\frac { d^2y }{dx^2}\) + x \(\frac { dy }{dx}\) + xy = 0

## Answer

Answer: (c) \(\frac { d^2y }{dx^2}\) – x² \(\frac { dy }{dx}\) + xy = 0

Question 7.

The general solution of the differential equation \(\frac { dy }{dx}\) = e^{x+y} is

(a) e^{x} + e^{-y} = c

(b) e^{x} + e^{y} = c

(c) e^{-x} + e^{y} = c

(d) e^{-x} + e^{-y} = c.

## Answer

Answer: (a) e^{x} + e^{-y} = c

Question 8.

Which of the following differential equations cannot be solved, using variable separable method?

(a) \(\frac { dy }{dx}\) + e^{x+y} + e^{-x+y}

(b) (y² – 2xy) dx = (x² – 2xy) dy

(c) xy \(\frac { dy }{dx}\) = 1 + x + y + xy

(d) \(\frac { dy }{dx}\) + y = 2.

## Answer

Answer: (b) (y² – 2xy) dx = (x² – 2xy) dy

Question 9.

A homogeneous differential equation of the form \(\frac { dy }{dx}\) = h(\(\frac { x }{y}\)) can be solved by making the substitution.

(a) y = vx

(b) v = yx

(c) x = vy

(d) x = v

## Answer

Answer: (c) x = vy

Question 10.

Which of the following is a homogeneous differential equation?

(a) (4x + 6y + 5)dy – (3y + 2x + 4)dx = 0

(b) xy dx – (x³ + y²)dy = Q

(c) (x³ + 2y²) dx + 2xy dy = 0

(d) y² dx + (x² – xy – y²)dy = 0.

## Answer

Answer: (d) y² dx + (x² – xy – y²)dy = 0.

Question 11.

The integrating factor of the differential equation x\(\frac { dy }{dx}\) – y = 2x² is

(a) e^{-x}

(b) e^{-y}

(c) \(\frac { 1 }{x}\)

(d) x

## Answer

Answer: (c) \(\frac { 1 }{x}\)

Question 12.

The integrating factor of the differential equation

(1 – y²) \(\frac { dy }{dx}\) + yx = ay(-1 < y < 1) is

(a) \(\frac { 1 }{y^2-1}\)

(b) \(\frac { 1 }{\sqrt{y^2-1}}\)

(c) \(\frac { 1 }{1-y^2}\)

(d) \(\frac { 1 }{\sqrt{1-y^2}}\)

## Answer

Answer: (d) \(\frac { 1 }{\sqrt{1-y^2}}\)

Question 13.

The general solution of the differential equation \(\frac { y dx – x dy }{y}\) = 0 is

(a) xy = c

(b) x = cy²

(c) y = cx

(d) y = cx².

## Answer

Answer: (c) y = cx

Question 14.

The general solution of a differential equation of the type \(\frac { dy }{dx}\) + P_{1} x = Q_{1} is:

(a) y e^{∫p1 dy} = ∫(Q_{1} e^{∫p1 dy}) dy + c

(b) y e^{∫p1 dx} = ∫(Q_{1} e^{∫p1 dx}) dx + c

(c) x e^{∫p1 dy} = ∫(Q_{1} e^{∫p1 dy}) dy + c

(d) x e^{∫p1 dx} = ∫(Q_{1} e^{∫p1 dx}) dx + c

## Answer

Answer: (c) x e^{∫p1 dy} = ∫(Q_{1} e^{∫p1 dy}) dy + c

Question 15.

The general solution of the differential equation

e^{x} dy + (y e^{x} + 2x) dx = 0 is

(a) x e^{x} + x² = c

(b) x e^{y} + y² = c

(c) y e^{x} + x² = c

(d) y e^{x} + x² = c.

## Answer

Answer: (c) y e^{x} + x² = c

Question 16.

The degree of the differential equation representing the family of curves (x – a)² + y² = 16 is

(a) 0

(b) 2

(c) 3

(d) 1.

## Answer

Answer: (d) 1.

Question 17.

The degree of the differential equation

\(\frac { d^2y }{dx^2}\) + 3(\(\frac { dy }{dx}\))² = x² log (\(\frac { d^2y }{dx^2}\)) is

(a) 1

(b) 2

(c) 3

(d) not defined

## Answer

Answer: (d) not defined

Question 18.

The order and degree of the differential equation

[1 + (\(\frac { dy }{dx}\))²]² = \(\frac { d^2y }{dx^2}\)

(a) 1, 2

(b) 2, 2

(c) 2, 1

(d) 4, 2.

## Answer

Answer: (c) 2, 1

Question 19.

The solution of the differential equation:

2x \(\frac { dy }{dx}\) – y = 3 represents a family of:

(a) straight lines

(b) circles

(c) parabolas

(d) ellipses.

## Answer

Answer: (c) parabolas

Question 20.

A solution of the differential equation:

(\(\frac { dy }{dx}\))² – x \(\frac { dy }{dx}\) + y = 0 is

(a) y = 2

(b) y = 2x

(c) y = 2x – 4

(d) y = 2x² – 4.

## Answer

Answer: (c) y = 2x – 4

Question 21.

The solution of the differential equation:

x\(\frac { dy }{dx}\) + 2y = x² is

(a) y = \(\frac { x^2+c }{4x^2}\)

(b) y = \(\frac { x^2 }{4}\) + c

(c) y = y = \(\frac { x^4+c }{x^2}\)

(d) y = y = \(\frac { x^4+c }{4x^2}\)

## Answer

Answer: (d) y = y = \(\frac { x^4+c }{4x^2}\)

Question 22.

The differential equation of the family of circles with fixed radius 5 units and centre on the line y = 2 is

(a) (x – 2)² y’² = 25 – (y – 2)²

(b) (x – 2) y’² = 25 – (y – 2)²

(c) (y – 2) y’² =25 – (y – 2)².

(d) (y – 2)² y’² = 25 – (y – 2)².

## Answer

Answer: (d) (y – 2)² y’² = 25 – (y – 2)².

Hint:

The equation of the circle is

(x – c)² + (y – 2)² = 25 …………… (1)

Diff. w.r.t. x,

2(x – c) + 2(y – 2)y’ = 0

⇒ (x – c) = -(y – 2)y’

Putting in (1),

(y – 2)² y’² + (y – 2)² = 25

(y – 2)² y’² = 25 – (y – 2)².

Question 23.

The differential equation which represents the family of curves y = e^{c2x}, where c_{1} and c_{2} are arbitrary constants, is:

(a) y” = y’y

(b) yy” = y’

(c) yy” = (y’)²

(d) y’ = y²

## Answer

Answer: (d) y’ = y²

Hint:

We have y = c_{1} e^{c2x} …………… (1)

Diff. w.r.t. x, y’ = c_{1}c_{2} e^{c2x}

⇒ y’ = c_{2}y ………… (2) [Using(1)]

Again diff. w.r.t. x,

y” = c_{2}y’ …………… (3)

From (2) and (3),

\(\frac { y” }{y’}\) = \(\frac { y’ }{y}\)

⇒ yy” = (y’)²

Question 24.

Solution of the differential equation:

cos x dy = y (sin x – y) dx, 0 < x < \(\frac { π }{2}\) is

(a) sec x = (tan x + c)y

(b) y sec x = tan x + c

(c) y tan x = sec x + c

(d) tan x = (sec x + c)y.

## Answer

Answer: (a) sec x = (tan x + c)y

Hint:

Here cos x dy =y (sin x – y)dx

⇒ cos x dy – y sin x dx = – y² dx

⇒ d(y cos x) = – y² dx.

Integrating, ∫\(\frac { d(y cos x) }{y^2 cos^2 x}\) = -∫\(\frac { 1 }{cos^2 x}\) dx

⇒ –\(\frac { 1 }{y cos x}\) = -tan x – c

⇒ sec x = (tan x + c)y.

Question 25.

At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production P w.r.t. additional number of worker x is given by:

\(\frac { dP }{dx}\) = 100 – 12√x

If the firm employs 25 more workers, then the new level of production of items is:

(a) 3000

(b) 3500

(c) 4500

(d) 2500.

## Answer

Answer: (b) 3500

Hint:

We have: \(\frac { dP }{dx}\) = 100 – 12√x

Integrating,

\(\int_{2000}^{P}\) dP = \(\int_{0}^{25}\) (100 – 12√x)dx

⇒ [P]\(_{2000}^{P}\) = [100x – 12\(\frac { x^{3/2} }{3/2}\)]\(_{0}^{25}\)

⇒ P – 2000 = 100(25)-8(25)^{3/2}

⇒ P – 2000 = 2500 – 1000

⇒ P = 3500.

Fill in the Blanks

Question 1.

The degree of the differential equation:

x²(\(\frac { d^2y }{dx^2}\))³ + y(\(\frac { dy }{dx}\))^{4} + x³ = 0 is …………….

## Answer

Answer: 3.

Question 2.

The degree and order of the differential equation:

(\(\frac { ds }{dt}\))^{4} + 3s\(\frac { d^2s }{dt^2}\) is ……………. and ………………..

## Answer

Answer: 2, 1

Question 3.

Differential equation of the family of lines passing through the origin is …………………

## Answer

Answer: \(\frac { dy }{dx}\) = \(\frac { y }{x}\).

Question 4.

The differential equation of which y = 2 (x² – 1) + ce^{-x} is a solution is ……………….

## Answer

Answer: \(\frac { dy }{dx}\) + 2xy = 4x³

Question 5.

General solution of (x² + 1)\(\frac { dy }{dx}\) = 2 is ………………….

## Answer

Answer: y = 2 tan^{-1} x + c.

Question 6.

Solution of \(\frac { dy }{dx}\) = \(\sqrt { 4 – y^2}\) (- 2 < y < 2) is …………….

## Answer

Answer: sin^{-1} \(\frac { y }{2}\) = x + c.

Question 7.

Solution of \(\frac { dy }{dx}\) = \(\frac { y }{x}\) is ……………….

## Answer

Answer: y = cx.

Question 8.

The differential equation \(\frac { dy }{dx}\) = \(\frac { x-y }{x+y}\) is ………………. equation.

## Answer

Answer: homogeneous.

Question 9.

The integrating factor of x log x \(\frac { dy }{dx}\) + y = 2 log x is …………………

## Answer

Answer: log x.

Question 10.

The integrating factor of (\(\frac { e^{-2√x} }{√x}\) – \(\frac { y }{√x}\))\(\frac { dy }{dx}\) = 1 is …………………

## Answer

Answer: e^{2√x}

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