# MCQ Questions for Class 12 Maths Chapter 9 Differential Equations with Answers

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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.

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.

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.

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.

Question 5.
Which of the following differential equations has y = c1 ex+ c2 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: (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: (c) $$\frac { d^2y }{dx^2}$$ – x² $$\frac { dy }{dx}$$ + xy = 0

Question 7.
The general solution of the differential equation $$\frac { dy }{dx}$$ = ex+y is
(a) ex + e-y = c
(b) ex + ey = c
(c) e-x + ey = c
(d) e-x + e-y = c.

Answer: (a) ex + e-y = c

Question 8.
Which of the following differential equations cannot be solved, using variable separable method?
(a) $$\frac { dy }{dx}$$ + ex+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: (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

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: (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: (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: (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.

Question 14.
The general solution of a differential equation of the type $$\frac { dy }{dx}$$ + P1 x = Q1 is:
(a) y e∫p1 dy = ∫(Q1 e∫p1 dy) dy + c
(b) y e∫p1 dx = ∫(Q1 e∫p1 dx) dx + c
(c) x e∫p1 dy = ∫(Q1 e∫p1 dy) dy + c
(d) x e∫p1 dx = ∫(Q1 e∫p1 dx) dx + c

Answer: (c) x e∫p1 dy = ∫(Q1 e∫p1 dy) dy + c

Question 15.
The general solution of the differential equation
ex dy + (y ex + 2x) dx = 0 is
(a) x ex + x² = c
(b) x ey + y² = c
(c) y ex + x² = c
(d) y ex + x² = c.

Answer: (c) y ex + 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.

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

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.

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.

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: (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: (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: (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 = ec2x, where c1 and c2 are arbitrary constants, is:
(a) y” = y’y
(b) yy” = y’
(c) yy” = (y’)²
(d) y’ = y²

Hint:
We have y = c1 ec2x …………… (1)
Diff. w.r.t. x, y’ = c1c2 ec2x
⇒ y’ = c2y ………… (2) [Using(1)]
Again diff. w.r.t. x,
y” = c2y’ …………… (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: (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.

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 …………….

Question 2.
The degree and order of the differential equation:
($$\frac { ds }{dt}$$)4 + 3s$$\frac { d^2s }{dt^2}$$ is ……………. and ………………..

Question 3.
Differential equation of the family of lines passing through the origin is …………………

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: $$\frac { dy }{dx}$$ + 2xy = 4x³

Question 5.
General solution of (x² + 1)$$\frac { dy }{dx}$$ = 2 is ………………….

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

Question 6.
Solution of $$\frac { dy }{dx}$$ = $$\sqrt { 4 – y^2}$$ (- 2 < y < 2) is …………….

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

Question 7.
Solution of $$\frac { dy }{dx}$$ = $$\frac { y }{x}$$ is ……………….

Question 8.
The differential equation $$\frac { dy }{dx}$$ = $$\frac { x-y }{x+y}$$ is ………………. equation.

Question 9.
The integrating factor of x log x $$\frac { dy }{dx}$$ + y = 2 log x is …………………

Question 10.
The integrating factor of ($$\frac { e^{-2√x} }{√x}$$ – $$\frac { y }{√x}$$)$$\frac { dy }{dx}$$ = 1 is …………………

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