JEE Mathematics Ordinary Differential Equations MCQs

Question. The order the differential equation \( \frac{d^2 y}{dx^2} - \left(\frac{dy}{dx}\right)^2 = 1 \) is
(a) one
(b) two
(c) four
(d) zero
Answer: (b) two

Question. The degree of the differential equation \( \sqrt{1 + \left(\frac{dy}{dx}\right)^2} = x^2 \) is
(a) one
(b) two
(c) half
(d) four
Answer: (b) two

Question. The differential equation of the family of curves \( y = e^x(A \cos x + B \sin x) \), where \( A, B \) are arbitrary constants, has the degree \( n \) and order \( m \). Then
(a) \( n = 2, m = 1 \)
(b) \( n = 2, m = 2 \)
(c) \( n = 1, m = 2 \)
(d) \( n = 1, m = 1 \)
Answer: (c) \( n = 1, m = 2 \)

Question. The general solution of a differential equation is \( y = ae^{bx+c} \) where \( a, b, c \) are arbitrary constants. The order of the differential equation is
(a) 3
(b) 2
(c) 1
(d) None of the options
Answer: (b) 2

Question. The general solution of a differential equation is \( (y + c)^2 = cx \) where \( c \) is an arbitrary constant. The order and degree of the differential equation are respectively
(a) 1, 2
(b) 2, 2
(c) 1, 1
(d) 2, 1
Answer: (a) 1, 2

Question. The degree and order of the differential equation of the family of all parabolas whose axis is the x-axis, are respectively
(a) 1, 2
(b) 3, 2
(c) 2, 3
(d) 2, 1
Answer: (a) 1, 2

Question. The order and degree of the differential equation of the family of circles touching the x-axis at the origin, are respectively
(a) 1, 1
(b) 1, 2
(c) 2, 1
(d) 2, 2
Answer: (a) 1, 1

Question. The order and degree of the differential equation of the family of ellipses having the same foci, are respectively
(a) 1, 1
(b) 2, 1
(c) 2, 2
(d) 1, 2
Answer: (d) 1, 2

Question. If \( y(t) \) is a solution of the equation \( (1 + t) \frac{dy}{dt} - ty = 1 \) and \( y(0) = -1 \) then \( y(1) \) is
(a) \( -\frac{1}{2} \)
(b) \( e + \frac{1}{2} \)
(c) \( e - \frac{1}{2} \)
(d) \( \frac{1}{2} \)
Answer: (a) \( -\frac{1}{2} \)

Question. The solution of \( (x + \log y)dy + y dx = 0 \) when \( y(0) = 1 \) is
(a) \( y(x - 1) + y\log y = 0 \)
(b) \( y(x - 1 + \log y) + 1 = 0 \)
(c) \( xy + y\log y + 1 = 0 \)
(d) None of the options
Answer: (b) \( y(x - 1 + \log y) + 1 = 0 \)

Question. The general solution of the equation \( (1 + y^2) + (x - e^{\tan^{-1} y}) \frac{dy}{dx} = 0 \) is
(a) \( 2x e^{\tan^{-1} y} = e^{2\tan^{-1} y} + k \)
(b) \( x e^{\tan^{-1} y} = \tan^{-1} y + k \)
(c) \( x e^{2\tan^{-1} y} = e^{\tan^{-1} y} + k \)
(d) \( x = 2 + ke^{-\tan^{-1} y} \)
Answer: (a) \( 2x e^{\tan^{-1} y} = e^{2\tan^{-1} y} + k \)

Question. Let \( \frac{df(x)}{dx} = \frac{e^{\sin x}}{x}, x > 0 \). If \( \int_1^4 \frac{3e^{\sin x^3}}{x} dx = f(k) - f(1) \) then one of the possible values of k is
(a) 16
(b) 63
(c) 64
(d) 15
Answer: (c) 64

Question. If \( x \frac{dy}{dx} + y = x \cdot \frac{f(xy)}{f'(xy)} \) then \( f(x \cdot y) \) is equal to (k being an arbitrary constant)
(a) \( ke^{x^2/2} \)
(b) \( ke^{y^3/2} \)
(c) \( ke^{xy/2} \)
(d) None of the options
Answer: (a) \( ke^{x^2/2} \)

Question. The differential equation \( \phi(x)dy = y \{\phi'(x) - y\}dx \) is changed in the form \( df(x, y) = 0 \). Then \( f(x, y) \) is
(a) \( \frac{1}{2}\phi(x) + y \)
(b) \( \frac{1}{y}\phi(x) - x \)
(c) \( \frac{1}{y}\phi(x) + x \)
(d) \( \frac{\phi(x)}{y} \)
Answer: (b) \( \frac{1}{y}\phi(x) - x \)

Question. The solution of primitive integral equation \( (x^2 + y^2)dy = xy \cdot dx \) is \( y = y(x) \). If \( y(1) = 1 \) and \( y(x_0) = e \) then \( x_0 \) is
(a) \( \sqrt{2(e^2 - 1)} \)
(b) \( \sqrt{2(e^2 + 1)} \)
(c) \( \sqrt{3}e \)
(d) \( \sqrt{\frac{1}{2}(e^2 + 1)} \)
Answer: (c) \( \sqrt{3}e \)