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

Question 1.
Integration factor of differential equation $$\frac{dy}{dx}$$ + py = Q, where P and IQ are functions of x is
(a) ∫epdx
(b) $$_{e}$$∫pdx
(c) $$_{e}$$-∫pdx
(d) None of these

Question 2.
The radius of a circle is increasing at the rate of 0.4 cm/ s. The rate of increasing of its circumference is
(a) 0.4 π cm/s
(b) 0.8 π cm/s
(c) 0.8 cm/s
(d) None of these

Question 3.
The solution of $$\frac{dy}{dx}$$ = 1 + x + y + xy is
(a) x – y = k(1 + xy)
(b) log (1 + y) = x + $$\frac{x^2}{2}$$ + k
(c) log (1 + x) + y + $$\frac{y^2}{2}$$ = k
(d) None of these

Answer: (b) log (1 + y) = x + $$\frac{x^2}{2}$$ + k

Question 4.
The degree of the differential equation
($$\frac{d^2y}{dx}$$)² + ($$\frac{dy}{dx}$$)² = x sin $$\frac{dy}{dx}$$ is
(a) 1
(b) 2
(c) 3
(d) not defined

Question 5.
The degree of differential equation
[1 + ($$\frac{dy}{dx}$$)²]$$\frac{3}{2}$$ = $$\frac{d^2y}{dx^2}$$ is
(a) 4
(b) $$\frac{3}{2}$$
(c) 2
(d) not defined

Question 6.
The order and degree of the differential equation
$$\frac{d^2y}{dx^2}$$ + ($$\frac{dy}{dx}$$)$$\frac{1}{4}$$ + x$$\frac{1}{3}$$ = 0 respectvely, are
(a) 2 and not defined
(b) 2 and 2
(c) 2 and 3
(d) 3 and 3

Answer: (a) 2 and not defined

Question 7.
If y = e-x (A cos x + B sin x), then y is a solution of
(a) $$\frac{d^2y}{dx^2}$$ + 2$$\frac{dy}{dx}$$ = 0
(b) $$\frac{d^2y}{dx^2}$$ – 2$$\frac{dy}{dx}$$ + 2y = 0
(c) $$\frac{d^2y}{dx^2}$$ + 2$$\frac{dy}{dx}$$ + 2y = 0
(d) $$\frac{d^2y}{dx^2}$$ + 2y = 0

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

Question 8.
The differential equation for y = A cos αx + B sin αx where A and B are arbitary constants is
(a) $$\frac{d^2y}{dx^2}$$ – α²y = 0
(b) $$\frac{d^2y}{dx^2}$$ + α²y = 0
(c) $$\frac{d^2y}{dx^2}$$ + αy = 0
(d) $$\frac{d^2y}{dx^2}$$ – αy = 0

Answer: (b) $$\frac{d^2y}{dx^2}$$ + α²y = 0

Question 9.
Solution of differential equation xdy – ydx = Q represents
(a) a rectangular hyperbola
(b) parabola whose vertex is at origin
(c) straight line passing through origin
(d) a circle whose centre is at origin

Answer: (c) straight line passing through origin

Question 10.
Integrating factor of the differential equation cos x $$\frac{dy}{dx}$$ + y sin x = 1 is
(a) cos x
(b) tan x
(c) sec x
(d) sin x

Question 11.
Solution of the differential equation tan y sec² x dx + tan x sec² y dy + 0 is .
(a) tan x + tan y = k
(b) tan x – tan y = k
(c) $$\frac{tan x}{tan y}$$ = k
(d) tan x.tan y = k

Answer: (d) tan x.tan y = k

Question 12.
Family r = Ax + A³ of curves is represented by the differential equation of degree
(a) 1
(b) 2
(c) 3
(d) 4

Question 13.
Integrating factor of $$\frac{xdy}{dx}$$ – y = x4 – 3x is
(a) x
(b) log x
(c) $$\frac{1}{2}$$
(d) -x

Answer: (c) $$\frac{1}{2}$$

Question 14.
Solution of $$\frac{dy}{dx}$$ – y = 1 y(0) = 1 is given by
(a) xy = -ex
(b) xy = -e-x
(c) xy = -1
(d) y = 2ex – 1

Answer: (d) y = 2ex – 1

Question 15.
The number of solutions of $$\frac{dy}{dx}$$ = $$\frac{y+1}{x-1}$$ when y(1) = 2 is
(a) none
(b) one
(c) two
(d) infinite

Question 16.
Which of the following is a second order differential equation?
(a) (y’)² + x = y²
(b) y’y” + y = sin x
(c) y” + (y”)² + y = 0
(d) y’ = y²

Answer: (b) y’y” + y = sin x

Question 17.
Integrating factor of the differential equation
(1 – x²) $$\frac{dy}{dx}$$ – xy = 1 is
(a) -x
(b) $$\frac{x}{1+x^2}$$
(c) $$\sqrt{1-x^2}$$
(d) $$\frac{1}{2}$$ log(1 – x²)

Answer: (c) $$\sqrt{1-x^2}$$

Question 18.
tan-1 x + tan-1 y = c is the general solution of the differential equation
(a) $$\frac{dy}{dx}$$ = $$\frac{1+y^2}{1+x^2}$$
(b) $$\frac{dy}{dx}$$ = $$\frac{1+x^2}{1+y^2}$$
(c) (1 + x²)dy + (1 + y²)dx = 0
(d) (1 +x²2)dx+(1 + y²)dy = 0

Answer: (c) (1 + x²)dy + (1 + y²)dx = 0

Question 19.
The differential equation y $$\frac{dy}{dx}$$ + x = c represents
(a) Family of hyperbolas
(b) Family of parabolas
(c) Family of ellipses
(d) Family of circles

Question 20.
The general solution of ex cos y dx – ex sin y dy = 0 is
(a) ex cos y = k
(b) ex sin y = k
(c) ex = k cos y
(d) ex = k sin y

Answer: (a) ex cos y = k

Question 21.
The degree of the differential equation
$$\frac{d^2y}{dx^2}$$ + ($$\frac{dy}{dx}$$)³ + 6y5 = 0 is
(a) 1
(b) 2
(c) 3
(d) 5

Question 22.
The solution of $$\frac{dy}{dx}$$ + y = e-x, y (0) = 0 is
(a) y = ex(x – 1)
(b) y = xe-x
(c) y = xe-x + 1
(d) y = (x + 1 )e-x

Question 23.
Integrating factor of the differential equation $$\frac{dy}{dx}$$ + y tan x – sec x = 0 is
(a) cos x
(b) sec x
(c) ecos x
(d) esec x

Question 24.
The solution of the differential equation $$\frac{dy}{dx}$$ = $$\frac{1+y^2}{1+x^2}$$
(a) y = tan-1 x
(b) y – x = k(1 + xy)
(c) x = tan-1 y
(d) tan (xy) = k

Answer: (b) y – x = k(1 + xy)

Question 25.
The integrating factor of the differential equation $$\frac{dy}{dx}$$ + y = $$\frac{1+y}{x}$$ is
(a) $$\frac{x}{e^x}$$
(b) $$\frac{e^x}{x}$$
(c) xex
(d) ex

Answer: (b) $$\frac{e^x}{x}$$

Question 26.
y = aemx + be-mx satisfies which of the following differential equation?
(a) $$\frac{dy}{dx}$$ + my = 0
(b) $$\frac{dy}{dx}$$ – my = 0
(c) $$\frac{d^2y}{dx^2}$$ – m²y = 0
(d) $$\frac{d^2y}{dx^2}$$ +m²y = 0

Answer: (c) $$\frac{d^2y}{dx^2}$$ – m²y = 0

Question 27.
The solution of the differential equation cos x sin y dx + sin x cos y dy = 0 is
(a) $$\frac{sin x}{sin y}$$ = c
(b) sin x sin y = c
(c) sin x + sin y = z
(d) cos x cos y = c

Answer: (b) sin x sin y = c

Question 28.
The solution of x $$\frac{dy}{dx}$$ + y = ex is
(a) y = $$\frac{e^x}{x}$$ + $$\frac{k}{x}$$
(b) y = xex + cx
(c) y = xex + k
(d) x = $$\frac{e^vy}{y}$$ + $$\frac{k}{y}$$

Answer: (a) y = $$\frac{e^x}{x}$$ + $$\frac{k}{x}$$

Question 29.
The differential equation of the family of cuves x² + y² – 2ay = 0, where a is arbitrary constant is
(a) (x² – y²)$$\frac{dy}{dx}$$ = 2xy
(b) 2 (x² + y²)$$\frac{dy}{dx}$$ = xy
(c) 2(x² – y²)$$\frac{dy}{dx}$$ = xy
(d) (x² + y²) $$\frac{dy}{dx}$$ = 2xy

Answer: (a) (x² – y²)$$\frac{dy}{dx}$$ = 2xy

Question 30.
Family y = Ax + A³ of curves will correspond to a differential equation of order
(a) 3
(b) 2
(c) 1
(d) not finite

Question 31.
The general solution of $$\frac{dy}{dx}$$ = 2x ex2-y is
(a) ex2-y = c
(b) e-y + ex2 = c
(c) ey = ex2 + c
(d) ex2+y = c

Answer: (c) ey = ex2 + c

Question 32.
The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is
(a) an ellipse
(b) parabola
(c) circle
(d) rectangular hyperbola

Question 33.
The general solution of the differential equation $$\frac{dy}{dx}$$ = e$$\frac{x^2}{2}$$ + xy is
(a) y = ce$$\frac{-x^2}{2}$$
(b) y = ce$$\frac{x^2}{2}$$
(c) y = (x + c)e$$\frac{x^2}{2}$$
(d) y = (c – x)e$$\frac{x^2}{2}$$

Answer: (c) y = (x + c)e$$\frac{x^2}{2}$$

Question 34.
The solution of the equation (2y – 1) dx-(2x + 3)dy = 0 is
(a) $$\frac{2x-1}{2y+3}$$ = k
(b) $$\frac{2y+1}{2x-3}$$ = k
(c) $$\frac{2x+3}{2y-1}$$ = k
(d) $$\frac{2x-1}{2y-1}$$ = k

Answer: (c) $$\frac{2x+3}{2y-1}$$ = k

Question 35.
The differential equation for which y = a cos x + b sin x is a solution is
(a) $$\frac{d^2y}{dx^2}$$ + y = 0
(b) $$\frac{d^2y}{dx^2}$$ – y = 0
(c) $$\frac{d^2y}{dx^2}$$ + (a + b)y = 0
(d) $$\frac{d^2y}{dx^2}$$ + (a – b)y = 0

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

Question 36.
The solution of $$\frac{dy}{dx}$$ + y = e-x, y (0) = 0 is
(a) y = e-x (x – 1)
(b) y = xex
(c) y = xe-x + 1
(d) y = xe-x

Question 37.
The order and degree of the differential equation
($$\frac{d^2y}{dx^3}$$)² – 3$$\frac{d^2y}{dx^2}$$ + 2($$\frac{dy}{dx}$$)4 = y4 are
(a) 1, 4
(b) 3, 4
(c) 2, 4
(d) 3, 2

Question 38.
The order and degree of the differential equation
[1 + ($$\frac{dy}{dx}$$)²] = $$\frac{d^2y}{dx^2}$$ are
(a) 1, $$\frac{3}{2}$$
(b) 2, 3
(c) 2, 1
(d) 3, 4

Question 39.
The differential equation of the family of curves y² = 4a (x + a) is
(a) y² = 4$$\frac{dy}{dx}$$ (x + $$\frac{dy}{dx}$$)
(b) 2y$$\frac{dy}{dx}$$ = 4a
(c) y$$\frac{d^2y}{dx^2}$$ + ($$\frac{dy}{dx}$$)² = 0
(d) 2x$$\frac{dy}{dx}$$ + y($$\frac{dy}{dx}$$)² – y

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

Question 40.
Which of the following is the general solution of $$\frac{d^2y}{dx^2}$$ – 2$$\frac{dy}{dx}$$ + y = 0
(a) y = (Ax + B)ex
(b) y = (Ax + B)e-x
(c) y = Aex + Be-x
(d) y = A cos x + B sin x