Differential Equations — Part 2 MCQs with Explanations

x2 + y2 = 4 is the correct answer to “Solution of $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = - \frac{{\text{x}}}{{\text{y}}}$$ at x = 1 and y = √3 is”. This question is standard in competitive examination syllabi for Differential Equations.

y = c1e-3x + c2e2x is the correct answer to “The solution to the ordinary differential equation $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + \frac{{{\text{dy}}}}{{{\text{dx}}}} - 6{\text{y}} = 0$$ is”. This question is standard in competitive examination syllabi for Differential Equations.

y = tan(x + 3) is the correct answer to “Consider the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = 1 + {{\text{y}}^2}.$$ Which one of the following can be a particular solution of this differential equation”. This question is standard in competitive examination syllabi for Differential Equations.

$$\frac{{{\text{dy}}}}{{{\text{dx}}}} + {\text{xy}} = {{\text{e}}^{ - {\text{x}}}}$$ is the correct answer to “Which ONE of the following is a linear non-homogeneous differential equation, where x and y are the independent and dependent variables respectively”. This question is standard in competitive examination syllabi for Differential Equations.

31.5°C is the correct answer to “A body originally at 60°C cools down to 40°C in 15 minutes when kept in air at a temperature of 25°C. What will be the temperature of the body at the end of 30 minutes”. This question is standard in competitive examination syllabi for Differential Equations.

$$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \left| {\text{x}} \right|$$ is the correct answer to “The figure shows the plot of y as a function of x The function shown is the solution of the differential equation (assuming all initial conditions to be zero) is”. This question is standard in competitive examination syllabi for Differential Equations.

degree = 1, order = 2 is the correct answer to “The following differential equation has $$3\left( {\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{t}}^2}}}} \right) + 4{\left( {\frac{{{\text{dy}}}}{{{\text{dt}}}}} \right)^3} + {{\text{y}}^{\text{2}}} + 2 = {\text{x}}$$”. This question is standard in competitive examination syllabi for Differential Equations.

1, 4, 9, ..... is the correct answer to “The boundary-value problem y'' + λy = 0, y(0) = y(π) = 0 will have non-zero solutions if and only if the values of λ are”. This question is standard in competitive examination syllabi for Differential Equations.

tan y + cot x = c (c is a constant) is the correct answer to “The general solution of the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \frac{{1 + \cos 2{\text{y}}}}{{1 - \cos 2{\text{x}}}}$$ is”. This question is standard in competitive examination syllabi for Differential Equations.

(1 + x)e-x2 is the correct answer to “The solution of the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} + 2{\text{xy}} = {{\text{e}}^{ - {{\text{x}}^2}}}$$ with y(0) = 1 is”. This question is standard in competitive examination syllabi for Differential Equations.

$${{\text{e}}^{ - {\text{x}}}}\left( {\cos 4{\text{x}} + \frac{1}{4}\sin 4{\text{x}}} \right)$$ is the correct answer to “The solution of $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + 2\frac{{{\text{dy}}}}{{{\text{dx}}}} + 17{\text{y}} = 0;$$ y(0) = 1, $$\frac{{{\text{dy}}}}{{{\text{dx}}}}\left( {\frac{\pi }{4}} \right) = 0$$ in the range 0 < x < $$\frac{\pi }{4}$$ is given by”. This question is standard in competitive examination syllabi for Differential Equations.

no solution is the correct answer to “The differential equation $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + 16{\text{y}} = 0$$ for y(x) with the two boundary conditions $${\left. {\frac{{{\text{dy}}}}{{{\text{dx}}}}} \right|_{{\text{x}} = 0}} = 1$$ and $${\left. {\frac{{{\text{dy}}}}{{{\text{dx}}}}} \right|_{{\text{x}} = \frac{\pi }{2}}} = - 1$$ has”. This question is standard in competitive examination syllabi for Differential Equations.

y = (c1 + c2log x) x + log x + 2 is the correct answer to “The solution of the differential equation $${{\text{x}}^2}\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} - {\text{x}}\frac{{{\text{dy}}}}{{{\text{dx}}}} + {\text{y}} = \log {\text{x}}$$ is”. This question is standard in competitive examination syllabi for Differential Equations.

ellipses is the correct answer to “Solution of the differential equation $$3{\text{y}}\frac{{{\text{dy}}}}{{{\text{dx}}}} + 2{\text{x}} = 0$$ represents a family of”. This question is standard in competitive examination syllabi for Differential Equations.

$${\text{y}} = \tan \left( {{\text{x}} + \frac{\pi }{4}} \right)$$ is the correct answer to “The solution of differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} - {{\text{y}}^2} = 1$$ satisfying condition y = 0 is”. This question is standard in competitive examination syllabi for Differential Equations.

2e-x2 is the correct answer to “The solution of the initial value problem $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = - 2{\text{xy}};$$ y(0) = 2 is”. This question is standard in competitive examination syllabi for Differential Equations.

y = Klnx is the correct answer to “An ordinary differential equation is given below. $$\left( {\frac{{{\text{dy}}}}{{{\text{dx}}}}} \right)\left( {{\text{x}}\ln {\text{x}}} \right) = {\text{y}}$$ The solution for the above equation is (Note: K denotes a constant in the options)”. This question is standard in competitive examination syllabi for Differential Equations.

$${\text{u}} = {\text{U}}\left( {\frac{{1 - {{\text{e}}^{{\text{kx}}}}}}{{1 - {{\text{e}}^{{\text{kL}}}}}}} \right)$$ is the correct answer to “The solution to the differential equation $$\frac{{{{\text{d}}^2} \cdot {\text{u}}}}{{{\text{d}}{{\text{x}}^2}}} - {\text{k}}\frac{{{\text{du}}}}{{{\text{dx}}}} = 0$$ is where k is constant, subjected to the boundary conditions u(0) = 0 and u(L) = U, is”. This question is standard in competitive examination syllabi for Differential Equations.

\[\frac{{\text{d}}}{{{\text{dt}}}}\left\{ {\begin{array}{*{20}{c}} {\text{x}} \\ {\text{y}} \end{array}} \right\} = \left[ {\begin{array}{*{20}{c}} 3&{ - 5} \\ 4&8 \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {\text{x}} \\ {\text{y}} \end{array}} \right\}\] is the correct answer to “The matrix form of the linear system $$\frac{{{\text{dx}}}}{{{\text{dt}}}} = 3{\text{x}} - 5{\text{y}}$$ and $$\frac{{{\text{dy}}}}{{{\text{dt}}}} = 4{\text{x}} + 8{\text{y}}$$ is”. This question is standard in competitive examination syllabi for Differential Equations.

Q(t) = 1 - e-t is the correct answer to “The solution of the equation $$\frac{{{\text{dQ}}}}{{{\text{dt}}}} + {\text{Q}} = 1$$ with Q = 0 at t = 0 is”. This question is standard in competitive examination syllabi for Differential Equations.

K1e(-1 + √6)x + K2e(-1 - √6)x is the correct answer to “The general solution of the differential equation $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + 2\frac{{{\text{dy}}}}{{{\text{dx}}}} - 5{\text{y}} = 0$$ in terms of arbitrary constants K 1 and K 2 is”. This question is standard in competitive examination syllabi for Differential Equations.

y = (c1 + c2ln x) x4 is the correct answer to “General solution of the Cauchy-Euler equation $${{\text{x}}^2}\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} - 7{\text{x}}\frac{{{\text{dy}}}}{{{\text{dx}}}} + 16{\text{y}} = 0$$ is”. This question is standard in competitive examination syllabi for Differential Equations.

$$\frac{1}{{\text{x}}} = \frac{1}{{\text{a}}} + {\text{kt}}$$ is the correct answer to “Biotransformation of an organic compound having concentration (x) can be modeled using an ordinary differential equation $$\frac{{{\text{dx}}}}{{{\text{dt}}}} + {\text{k}}{{\text{x}}^2} = 0,$$ where k is the reaction rate constant. If x = a at t = 0, the solution of the equation is”. This question is standard in competitive examination syllabi for Differential Equations.

u = 6e-(3x + 2t) is the correct answer to “The solution of initial value problem; $$\frac{{\partial {\text{u}}}}{{\partial {\text{x}}}} = 2\frac{{\partial {\text{u}}}}{{\partial {\text{t}}}} + {\text{u,}}$$ where u(x, 0) = 6e -3x is”. This question is standard in competitive examination syllabi for Differential Equations.

n(x) = K exp(-x/L) is the correct answer to “A function n(x) satisfies the differential equation $$\frac{{{{\text{d}}^2}{\text{n}}\left( {\text{x}} \right)}}{{{\text{d}}{{\text{x}}^2}}} - \frac{{{\text{n}}\left( {\text{x}} \right)}}{{{{\text{L}}^2}}} = 0$$ where L is a constant. The boundary conditions are: n(0) = K and n($$\infty $$) = 0. The solution to this equation is”. This question is standard in competitive examination syllabi for Differential Equations.

e-(1 + i)x, e(1 - i)x is the correct answer to “The solutions of differential equation $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + \frac{{{\text{2dy}}}}{{{\text{dx}}}} + 2{\text{y}} = 0$$ are”. This question is standard in competitive examination syllabi for Differential Equations.

0 is the correct answer to “It is given that y'' + 2y' + y = 0, y(0) = 0, y(1) = 0. What is y (0.5)”. This question is standard in competitive examination syllabi for Differential Equations.

5e-2 is the correct answer to “A function y(t), such that y(0) = 1 and y(1) = 3e -1 , is a solution of the differential equation $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{t}}^2}}} + 2\frac{{{\text{dy}}}}{{{\text{dt}}}} + {\text{y}} = 0.$$ Then y(2) is”. This question is standard in competitive examination syllabi for Differential Equations.

$$\ln \left( {1 + \frac{{{{\text{y}}^2}}}{{{{\text{x}}^2}}}} \right) = {\text{x}} - 1$$ is the correct answer to “A curve passes through the point (x = 1, y = 0) and satisfies the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \frac{{{{\text{x}}^2} + {{\text{y}}^2}}}{{2{\text{y}}}} + \frac{{\text{y}}}{{\text{x}}}.$$ The equation that describes the curve is”. This question is standard in competitive examination syllabi for Differential Equations.

y = e-4x + 1.25 is the correct answer to “The differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} + 4{\text{y}} = 5$$ is valid in the domain 0 ≤ x ≤ 1 with y(0) = 2.25. The solution of differential equation is”. This question is standard in competitive examination syllabi for Differential Equations.

1 is the correct answer to “The degree of the differential equation $$\frac{{{{\text{d}}^2}{\text{x}}}}{{{\text{d}}{{\text{t}}^2}}} + 2{{\text{x}}^3} = 0$$ is”. This question is standard in competitive examination syllabi for Differential Equations.

$$\frac{1}{3}\sin 3{\text{t}} + \cos 3{\text{t}}$$ is the correct answer to “The solution for the differential equation $$\frac{{{{\text{d}}^2}{\text{x}}}}{{{\text{d}}{{\text{t}}^2}}} = - 9{\text{x}}$$ with initial conditions x(0) = 1 and $${\left. {\frac{{{\text{dx}}}}{{{\text{dt}}}}} \right|_{{\text{t}} = 0}} = 1,$$ is”. This question is standard in competitive examination syllabi for Differential Equations.

$${\text{y}} = \tan \left( {\frac{{{{\text{x}}^2}}}{2} + {\text{c}}} \right)$$ is the correct answer to “Consider the differential equation: $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \left( {1 + {{\text{y}}^2}} \right){\text{x}}{\text{.}}$$ The general solution with constant c is”. This question is standard in competitive examination syllabi for Differential Equations.

6 months is the correct answer to “A spherical naphthalene ball exposed to the atmosphere loses volume at a rate proportional to its instantaneous surface area due to evaporation. If the initial diameter of the ball is 2 cm and the diameter reduces to 1 cm after 3 months, the ball completely evaporates in”. This question is standard in competitive examination syllabi for Differential Equations.

(3 - 18x)e-6x is the correct answer to “The particular solution of the initial value problem given below is $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + 12\frac{{{\text{dy}}}}{{{\text{dx}}}} + 36{\text{y}} = 0$$ with y(0) = 3 and $${\left. {\frac{{{\text{dy}}}}{{{\text{dx}}}}} \right|_{{\text{x}} = 0}} = - 36$$”. This question is standard in competitive examination syllabi for Differential Equations.

$${\text{y}} = {{\text{c}}_1}{\text{x}} + \frac{{{{\text{c}}_2}}}{{\text{x}}}$$ is the correct answer to “The solution to x 2 y'' + y' - y = 0 is”. This question is standard in competitive examination syllabi for Differential Equations.

$${\text{xy}}\cos \frac{{\text{y}}}{{\text{x}}} = {\text{c}}$$ is the correct answer to “Consider the following difference equation $${\text{x}}\left( {{\text{ydx}} + {\text{xdy}}} \right)\cos \frac{{\text{y}}}{{\text{x}}} = {\text{y}}\left( {{\text{xdy}} - {\text{ydx}}} \right)\sin \frac{{\text{y}}}{{\text{x}}}$$ Which of the following is the solution of the above equation (c is an arbitrary constant)”. This question is standard in competitive examination syllabi for Differential Equations.

$$\frac{1}{{\text{x}}}$$ is the correct answer to “Consider the differential equation $${{\text{x}}^2}\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + {\text{x}}\frac{{{\text{dy}}}}{{{\text{dx}}}} - {\text{y}} = 0.$$ Which of the following is a solution to this differential equation for x > 0”. This question is standard in competitive examination syllabi for Differential Equations.

$${\text{z}} = {\text{ax}} + {\text{by}} + \root 3 \of {{\text{ab}} + 2{{\left( {{{\text{a}}^2} + {\text{b}}} \right)}^2}} $$ is the correct answer to “The complete integral of (z - px - qy) 3 = pq + 2(p 2 + q) 2 is”. This question is standard in competitive examination syllabi for Differential Equations.

ek2t is the correct answer to “The integrating factor for differential equation $$\frac{{{\text{dP}}}}{{{\text{dt}}}} + {{\text{k}}_2}{\text{P}} = {{\text{k}}_1}{{\text{L}}_0}{{\text{e}}^{ - {{\text{k}}_1}{\text{t}}}}$$ is”. This question is standard in competitive examination syllabi for Differential Equations.