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

Answer

Answer: (d) None of these

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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

Answer

Answer: (b) 0.8 π cm/s

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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

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

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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

Answer

Answer: (d) not defined

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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

Answer

Answer: (c) 2

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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

Answer: (a) 2 and not defined

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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

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

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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

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

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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

Answer: (c) straight line passing through origin

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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

Answer

Answer: (c) sec x

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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

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

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

Answer

Answer: (b) 2

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

Answer

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

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

Answer

Answer: (d) y = 2e^x – 1

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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

Answer

Answer: (b) one

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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

Answer: (b) y’y” + y = sin x

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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

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

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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

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

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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

Answer

Answer: (d) Family of circles

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

Answer

Answer: (a) e^x cos y = k

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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

Answer

Answer: (a) 1

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

Answer

Answer: (b) y = xe^-x

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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) e^{cos x}
(d) e^{sec x}

Answer

Answer: (b) sec x

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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

Answer: (b) y – x = k(1 + xy)

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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) xe^x
(d) e^x

Answer

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

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Question 26.
y = ae^mx + 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

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

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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

Answer: (b) sin x sin y = c

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

Answer

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

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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

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

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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

Answer

Answer: (b) 2

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Question 31.
The general solution of \(\frac{dy}{dx}\) = 2x e^x2-y is
(a) e^x2-y = c
(b) e^-y + e^x2 = c
(c) e^y = e^x2 + c
(d) e^x2+y = c

Answer

Answer: (c) e^y = e^x2 + c

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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

Answer

Answer: (d) rectangular hyperbola

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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

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

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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

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

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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

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

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

Answer

Answer: (d) y = xe^-x

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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

Answer

Answer: (d) 3, 2

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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

Answer

Answer: (c) 2, 1

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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

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

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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)e^x
(b) y = (Ax + B)e^-x
(c) y = Ae^x + Be^-x
(d) y = A cos x + B sin x

Answer

Answer: (a) y = (Ax + B)e^x

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