JEE Mathematics Differential Equations MCQs Set 05

Question. The order and degree of the differential equation \( \left( 1+3 \frac{dy}{dx} \right)^{2/3} = 4 \frac{d^3y}{dx^3} \) are
(a) \( 1, \frac{2}{3} \)
(b) 3, 1
(c) 1, 2
(d) 3, 3
Answer: (d) 3, 3

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

Question. The order and degree of the differential equation \( \sqrt[3]{\frac{dy}{dx}} - 4 \frac{d^2y}{dx^2} - 7x = 0 \) are a and b, then a + b is
(a) 3
(b) 4
(c) 5
(d) 6
Answer: (c) 5

Question. Number of values of \( m \in N \) for which \( y = e^{mx} \) is a solution of the differential equation \( D^3y - 3D^2y - 4Dy + 12y = 0 \) is
(a) 0
(b) 1
(c) 2
(d) more than 2
Answer: (c) 2

Question. The value of the constant 'm' and 'c' for which \( y = mx + c \) is a solution of the differential equation \( D^2y - 3Dy - 4y = -4x \)
(a) is m = - 1, c = 3/4
(b) is m = 1, c = 3/4
(c) no such real m, c
(d) is m = 1, c = -3/4
Answer: (d) is m = 1, c = -3/4

Question. The differential equation of the family of curves represented by \( y = a + bx + ce^{-x} \) (where a, b, c are arbitrary constants) is
(a) \( y''' = y' \)
(b) \( y''' + y'' = 0 \)
(c) \( y''' - y'' + y' = 0 \)
(d) \( y''' + y'' - y' = 0 \)
Answer: (b) \( y''' + y'' = 0 \)

Question. The differential equation whose solution is \( Ax^2 + By^2 = 1 \), where A and B are arbitrary constants is of-
(a) first order and first degree
(b) second order and first degree
(c) second order and second degree
(d) first order and second degree
Answer: (b) second order and first degree

Question. The differential equation whose solution is \( (x - h)^2 + (y - k)^2 = a^2 \) is (where a is a constant)
(a) \( \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^3 = a^2 \left( \frac{d^2y}{dx^2} \right)^2 \)
(b) \( \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^3 = a^2 \frac{d^2y}{dx^2} \)
(c) \( \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^3 = a^2 \left( \frac{d^2y}{dx^2} \right)^2 \)
(d) None of the options
Answer: (a) \( \left[ 1 + \left( \frac{dy}{dx} \right)^2 \right]^3 = a^2 \left( \frac{d^2y}{dx^2} \right)^2 \)

Question. If \( y = e^{(K+1)x} \) is a solution of differential equation \( \frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 4y = 0 \), then k equals
(a) -1
(b) 0
(c) 1
(d) 2
Answer: (c) 1

Question. The differential equation for the family of curves \( x^2 + y^2 - 2ay = 0 \), where a is an arbitrary constant is
(a) \( (x^2 - y^2)y' = 2xy \)
(b) \( 2(x^2 + y^2)y' = xy \)
(c) \( 2(x^2 - y^2)y' = xy \)
(d) \( (x^2 + y^2)y' = 2xy \)
Answer: (a) \( (x^2 - y^2)y' = 2xy \)

Question. The solution to the differential equation \( y\ln y + xy' = 0 \), where \( y(1) = e \), is
(a) \( x(\ln y) = 1 \)
(b) \( xy(\ln y) = 1 \)
(c) \( (\ln y)^2 = 2 \)
(d) \( \ln y + \frac{x^2}{2} = 1 \)
Answer: (a) \( x(\ln y) = 1 \)

Question. A curve passing through (2, 3) and satisfying the differential equation \( \int_0^x ty(t)dt = x^2 y(x), (x > 0) \) is
(a) \( x^2 + y^2 = 13 \)
(b) \( y^2 = \frac{9}{2} x \)
(c) \( \frac{x^2}{8} + \frac{y^2}{18} = 1 \)
(d) \( xy = 6 \)
Answer: (d) xy = 6

Question. The equation of the curve passing through origin and satisfying the differential equation \( \frac{dy}{dx} = \sin(10x + 6y) \) is
(a) \( y = \frac{1}{3} \tan^{-1} \left( \frac{5 \tan 4x}{4 - 3 \tan 4x} \right) - \frac{5x}{3} \)
(b) \( y = \frac{1}{3} \tan^{-1} \left( \frac{5 \tan 4x}{4 + 3 \tan 4x} \right) - \frac{5x}{3} \)
(c) \( y = \frac{1}{3} \tan^{-1} \left( \frac{3 + \tan 4x}{4 - 3 \tan 4x} \right) - \frac{5x}{3} \)
(d) None of the options
Answer: (a) \( y = \frac{1}{3} \tan^{-1} \left( \frac{5 \tan 4x}{4 - 3 \tan 4x} \right) - \frac{5x}{3} \)

Question. If \( x \frac{dy}{dx} = y(\log y - \log x + 1) \), then the solution of the equation is
(a) \( \log \left( \frac{x}{y} \right) = cy \)
(b) \( \log \left( \frac{y}{x} \right) = cx \)
(c) \( x \log \left( \frac{y}{x} \right) = cy \)
(d) \( y \log \left( \frac{x}{y} \right) = cx \)
Answer: (b) \( \log \left( \frac{y}{x} \right) = cx \)

Question. The solution of the differential equation \( (2x - 10y^3) \frac{dy}{dx} + y = 0 \) is
(a) \( x + y = ce^{2x} \)
(b) \( y^2 = 2x^3 + c \)
(c) \( xy^2 = 2y^5 + c \)
(d) \( x(y^2 + xy) = 0 \)
Answer: (c) \( xy^2 = 2y^5 + c \)

Question. Solution of differential equation \( (1 + y^2)dx + (x - e^{-\tan^{-1}y})dy = 0 \) is
(a) \( y e^{\tan^{-1}x} = \tan^{-1}x + c \)
(b) \( x e^{\tan^{-1}y} = \frac{1}{2} e^{2\tan^{-1}y} + c \)
(c) \( 2x = e^{\tan^{-1}y} + c \)
(d) \( y = x e^{-\tan^{-1}x} + c \)
Answer: (b) \( x e^{\tan^{-1}y} = \frac{1}{2} e^{2\tan^{-1}y} + c \)

Question. The general solution of the differential equation, \( y' + y\phi'(x) - \phi(x)\phi'(x) = 0 \) where \( \phi(x) \) is a known function is
(a) \( y = ce^{-\phi(x)} + \phi(x) - 1 \)
(b) \( y = ce^{\phi(x)} + \phi(x) + K \)
(c) \( y = ce^{\phi(x)} - \phi(x) + 1 \)
(d) \( y = ce^{-\phi(x)} + \phi(x) + K \)
Answer: (a) \( y = ce^{-\phi(x)} + \phi(x) - 1 \)

Question. The solution of the differential equation, \( e^x(x+1)dx + (ye^y - xe^x)dy = 0 \) with initial condition f(0) = 0, is
(a) \( xe^x + 2y^2 e^y = 0 \)
(b) \( 2xe^x + y^2 e^y = 0 \)
(c) \( xe^x - 2y^2 e^y = 0 \)
(d) \( 2xe^x - y^2 e^y = 0 \)
Answer: (b) \( 2xe^x + y^2 e^y = 0 \)

Question. The solution of \( y^5x + y - x \frac{dy}{dx} = 0 \) is
(a) \( x^4/4 + 1/5(x/y)^5 = C \)
(b) \( x^5/5 + (1/4)(x/y)^4 = C \)
(c) \( (x/y)^5 + x^4/4 = C \)
(d) \( (xy)^4 + x^5/5 = C \)
Answer: (b) \( x^5/5 + (1/4)(x/y)^4 = C \)

Question. The solution of \( \frac{xdy}{x^2+y^2} = \left( \frac{y}{x^2+y^2} - 1 \right) dx \) is
(a) \( y = x \cot(c-x) \)
(b) \( \cos^{-1}(y/x) = -x + c \)
(c) \( y = x \tan(c-x) \)
(d) \( y^2/x^2 = x \tan(c-x) \)
Answer: (c) \( y = x \tan(c-x) \)

Question. Which one of the following curves represents the solution of the initial value problem \( Dy = 100 - y \) where \( y(0) = 50 \)?
(a) [Graph showing y starting at 50 and decreasing toward 0]
(b) [Graph showing y starting at 50 and increasing asymptotically toward 100]
(c) [Graph showing y starting at 50 and increasing exponentially]
(d) [Graph showing y starting at 50 and increasing linearly]
Answer: (b) [Graph showing y starting at 50 and increasing asymptotically toward 100]

Question. A solution of the differential equation, \( \left(\frac{dy}{dx}\right)^2 - x \frac{dy}{dx} + y = 0 \) is
(a) \( y = 2 \)
(b) \( y = 2x \)
(c) \( y = 2x - 4 \)
(d) \( y = 2x^2 - 4 \)
Answer: (c) \( y = 2x - 4 \)

Question. The differential equation representing the family of curves, \( y^2 = 2c(x + \sqrt{c}) \), where \( c \) is a positive parameter, is of
(a) order 1
(b) order 2
(c) degree 3
(d) degree 4
Answer: (c) degree 3

Question. Which one of the following is homogeneous function?
(a) \( f(x, y) = \frac{x - y}{x^2 + y^2} \)
(b) \( f(x, y) = x^{1/3} \cdot y^{-2/3} \tan^{-1} \frac{x}{y} \)
(c) \( f(x, y) = x (\ln \sqrt{x^2 + y^2} - \ln y) + y e^{x/y} \)
(d) \( f(x, y) = x \left[ \ln \left( \frac{2x^2 + y^2}{x} \right) - \ln(x + y) \right] + y^2 \tan \left( \frac{x + 2y}{3x - y} \right) \)
Answer: (a) \( f(x, y) = \frac{x - y}{x^2 + y^2} \), (b) \( f(x, y) = x^{1/3} \cdot y^{-2/3} \tan^{-1} \frac{x}{y} \), (c) \( f(x, y) = x (\ln \sqrt{x^2 + y^2} - \ln y) + y e^{x/y} \)

Question. The function \( f(x) \) satisfying the equation, \( f^2(x) + 4f'(x) \cdot f(x) + [f'(x)]^2 = 0 \) is
(a) \( f(x) = c \cdot e^{(2 - \sqrt{3})x} \)
(b) \( f(x) = c \cdot e^{(2 + \sqrt{3})x} \)
(c) \( f(x) = c \cdot e^{(\sqrt{3} - 2)x} \)
(d) \( f(x) = c \cdot e^{-(2 + \sqrt{3})x} \)
Answer: (c) \( f(x) = c \cdot e^{(\sqrt{3} - 2)x} \), (d) \( f(x) = c \cdot e^{-(2 + \sqrt{3})x} \)

Question. The equation of the curve passing through (3, 4) & satisfying the differential equation, \( y \left( \frac{dy}{dx} \right)^2 + (x - y) \frac{dy}{dx} - x = 0 \) can be
(a) \( x - y + 1 = 0 \)
(b) \( x^2 + y^2 = 25 \)
(c) \( x^2 + y^2 - 5x - 10 = 0 \)
(d) \( x + y - 7 = 0 \)
Answer: (a) \( x - y + 1 = 0 \), (b) \( x^2 + y^2 = 25 \)

Question. The graph of the function \( y = f(x) \) passing through the point (0, 1) and satisfying the differential equation \( \frac{dy}{dx} + y \cos x = \cos x \) is such that
(a) it is a constant function
(b) it is periodic
(c) it is neither an even nor an odd function
(d) it is continuous & differentiable for all x.
Answer: (a) it is a constant function, (b) it is periodic, (d) it is continuous & differentiable for all x.

Question. Water is drained from a vertical cylindrical tank by opening a valve at the base of the tank. It is known that the rate at which the water level drops is proportional to the square root of water depth y, where the constant of proportionality \( k > 0 \) depends on the acceleration due to gravity and the geometry of the hole. If t is measured in minutes and \( k = 1/15 \) then the time to drain the tank if the water is 4 meter deep to start with is
(a) 30 min
(b) 45 min
(c) 60 min
(d) 80 min
Answer: (c) 60 min

Question. Number of straight lines which satisfy the differential equation \( \frac{dy}{dx} + x \left( \frac{dy}{dx} \right)^2 - y = 0 \) is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2

Question. The solution of the differential equation, \( x^2 \frac{dy}{dx} \cdot \cos \frac{1}{x} - y \sin \frac{1}{x} = -1 \), where \( y \to -1 \) as \( x \to \infty \) is
(a) \( y = \sin \frac{1}{x} - \cos \frac{1}{x} \)
(b) \( y = \frac{x + 1}{x \sin \frac{1}{x}} \)
(c) \( y = \cos \frac{1}{x} + \sin \frac{1}{x} \)
(d) \( y = \frac{x + 1}{x \cos \frac{1}{x}} \)
Answer: (a) \( y = \sin \frac{1}{x} - \cos \frac{1}{x} \)

Question. If \( y = \frac{x}{\ln |cx|} \) (where c is an arbitrary constant) is the general solution of the differential equation \( \frac{dy}{dx} = \frac{y}{x} + \phi \left( \frac{x}{y} \right) \) then the function \( \phi \left( \frac{x}{y} \right) \) is
(a) \( \frac{x^2}{y^2} \)
(b) \( -\frac{x^2}{y^2} \)
(c) \( \frac{y^2}{x^2} \)
(d) \( -\frac{y^2}{x^2} \)
Answer: (d) \( -\frac{y^2}{x^2} \)

Question. If \( \int_a^x t y(t) dt = x^2 + y(x) \) then y as a function of x is
(a) \( y = 2 - (2 + a^2) e^{\frac{x^2 - a^2}{2}} \)
(b) \( y = 1 - (2 + a^2) e^{\frac{x^2 - a^2}{2}} \)
(c) \( y = 2 - (1 + a^2) e^{\frac{x^2 - a^2}{2}} \)
(d) None of the options
Answer: (a) \( y = 2 - (2 + a^2) e^{\frac{x^2 - a^2}{2}} \)

Question. A function \( f(x) \) satisfying \( \int_0^1 f(tx) dt = n f(x) \), where \( x > 0 \), is
(a) \( f(x) = c \cdot x^{\frac{1-n}{n}} \)
(b) \( f(x) = c \cdot x^{\frac{n}{n-1}} \)
(c) \( f(x) = c \cdot x^{\frac{1}{n}} \)
(d) \( f(x) = c \cdot x^{(1-n)} \)
Answer: (a) \( f(x) = c \cdot x^{\frac{1-n}{n}} \)

Question. The differential equation \( \frac{d^2y}{dx^2} + \frac{dy}{dx} + \sin y + x^2 = 0 \) is of the following type
(a) linear
(b) homogeneous
(c) order two
(d) degree one
Answer: (c) order two, (d) degree one

Question. A curve C passes through origin and has the property that at each point (x, y) on it the normal line at that point passes through (1, 0). The equation of a common tangent to the curve C and the parabola \( y^2 = 4x \) is
(a) \( x = 0 \)
(b) \( y = 0 \)
(c) \( y = x + 1 \)
(d) \( x + y + 1 = 0 \)
Answer: (a) \( x = 0 \)

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

Question. Let \( y = (A + Bx) e^{3x} \) be a solution of the differential equation \( \frac{d^2y}{dx^2} + m \frac{dy}{dx} + ny = 0, m, n \in I \), then
(a) \( m + n = 3 \)
(b) \( n^2 - m^2 = 64 \)
(c) \( m = -6 \)
(d) \( n = 9 \)
Answer: (a) \( m + n = 3 \), (c) \( m = -6 \), (d) \( n = 9 \)

Question. The differential equation \( 2xy dy = (x^2 + y^2 + 1) dx \) determines
(a) A family of circles with centre on x-axis
(b) A family of circles with centre on y-axis
(c) A family of rectangular hyperbola with centre on x-axis
(d) A family of rectangular hyperbola with centre on y-axis
Answer: (c) A family of rectangular hyperbola with centre on x-axis

Question. If \( f''(x) + f'(x) + f^2(x) = x^2 \) be the differential equation of a curve and let P be the point of maxima then number of tangents which can be drawn from point P to \( x^2 - y^2 = a^2 \) is
(a) 2
(b) 1
(c) 0
(d) either 1 or 2
Answer: (d) either 1 or 2

Question. The solution of \( x^2 dy - y^2 dx + xy^2(x - y) dy = 0 \) is
(a) \( \ln \left| \frac{x - y}{xy} \right| = \frac{y^2}{2} + c \)
(b) \( \ln \left| \frac{xy}{x - y} \right| = \frac{x^2}{2} + c \)
(c) \( \ln \left| \frac{x - y}{xy} \right| = \frac{x^2}{2} + c \)
(d) \( \ln \left| \frac{x - y}{xy} \right| = x + c \)
Answer: (a) \( \ln \left| \frac{x - y}{xy} \right| = \frac{y^2}{2} + c \)

Question. The orthogonal trajectories of the system of curves \( \left( \frac{dy}{dx} \right)^2 = \frac{4}{x} \) are
(a) \( 9(y + c)^2 = x^3 \)
(b) \( y + c = \frac{-x^{3/2}}{3} \)
(c) \( y + c = \frac{x^{3/2}}{3} \)
(d) All of the options
Answer: (a) \( 9(y + c)^2 = x^3 \), (b) \( y + c = \frac{-x^{3/2}}{3} \), (c) \( y + c = \frac{x^{3/2}}{3} \), (d) All of the options