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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 = 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
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 \(\frac { dy }{dx}\) = ex+y is
(a) ex + e-y = c
(b) ex + ey = c
(c) e-x + ey = c
(d) e-x + e-y = c.
Answer
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
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 \(\frac { dy }{dx}\) + 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
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
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.
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 = ec2x, where c1 and c2 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 = 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
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: e2√x
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