Differential Equations — Part 1 MCQs with Explanations

$${\text{y}} = \left( {{{\text{C}}_1} + {{\text{C}}_2}{\text{x}}} \right){{\text{e}}^{\text{x}}} + {{\text{C}}_3}\cos {\text{x}} + {{\text{C}}_4}\sin {\text{x}}$$ is the correct answer to “The general solution of the differential equation, $$\frac{{{{\text{d}}^4}{\text{y}}}}{{{\text{d}}{{\text{x}}^4}}} - 2\frac{{{{\text{d}}^3}{\text{y}}}}{{{\text{d}}{{\text{x}}^3}}} + 2\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} - 2\frac{{{\text{dy}}}}{{{\text{dx}}}} + {\text{y}} = 0$$ is”. This question is standard in competitive examination syllabi for Differential Equations.

$$\sqrt {1 - {{\text{x}}^2}} + \sqrt {1 - {{\text{y}}^2}} = {\text{c}}$$ is the correct answer to “The solution of the differential equation, $${\text{y}}\sqrt {1 - {{\text{x}}^2}} {\text{dy}} + {\text{x}}\sqrt {1 - {{\text{y}}^2}} {\text{dx}} = {\text{0}}$$ is”. This question is standard in competitive examination syllabi for Differential Equations.

$$\sqrt 2 $$ is the correct answer to “The maximum value of the solution y(t) of the differential equation \[{\rm{y}}\left( {\rm{t}} \right) + {\rm{\ddot y}}\left( {\rm{t}} \right) = 0\] with initial conditions \[{\rm{\dot y}}\left( 0 \right) = 1\] and y(0) = 1, for t ≥ 0 is”. This question is standard in competitive examination syllabi for Differential Equations.

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

3 and 2 is the correct answer to “The order and degree of the differential equation $$\frac{{{{\text{d}}^3}{\text{y}}}}{{{\text{d}}{{\text{x}}^3}}} + 4\sqrt {{{\left( {\frac{{{\text{dy}}}}{{{\text{dx}}}}} \right)}^3} + {{\text{y}}^2}} = 0$$ are respectively”. This question is standard in competitive examination syllabi for Differential Equations.

-2 - 2t - t2 is the correct answer to “Consider the following second-order differential equation: y" - 4y' + 3y = 2t - 3t 2 The particular solution of the differential equation is”. This question is standard in competitive examination syllabi for Differential Equations.

2nd order and 2nd degree is the correct answer to “The differential equation $${\left[ {1 + {{\left( {\frac{{{\text{dy}}}}{{{\text{dx}}}}} \right)}^2}} \right]^3} = {{\text{C}}^2}{\left[ {\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}}} \right]^2}$$ is of”. This question is standard in competitive examination syllabi for Differential Equations.

$${{\text{x}}_{1{\text{f}}}} = {{\text{x}}_{2{\text{f}}}} < \infty $$ is the correct answer to “Consider a system governed by the following equations: $$\frac{{{\text{d}}{{\text{x}}_1}\left( {\text{t}} \right)}}{{{\text{dt}}}} = {{\text{x}}_2}\left( {\text{t}} \right) - {{\text{x}}_1}\left( {\text{t}} \right);\,\frac{{{\text{d}}{{\text{x}}_2}\left( {\text{t}} \right)}}{{{\text{dt}}}} = {{\text{x}}_1}\left( {\text{t}} \right) - {{\text{x}}_2}\left( {\text{t}} \right)$$ The initial conditions are such that $${{\text{x}}_1}\left( 0 \right) < {{\text{x}}_2}\left( 0 \right) < \infty .$$ Let $${{\text{x}}_{1{\text{f}}}} = \mathop {\lim }\limits_{{\text{t}} \to \infty } {{\text{x}}_1}\left( {\text{t}} \right)$$ and $${{\text{x}}_{2{\text{f}}}} = \mathop {\lim }\limits_{{\text{t}} \to \infty } {{\text{x}}_2}\left( {\text{t}} \right).$$ Which one of the following is true”. This question is standard in competitive examination syllabi for Differential Equations.

Hyperbolas and Circles is the correct answer to “The families of curves represented by the solution of the equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = - {\left( {\frac{{\text{x}}}{{\text{y}}}} \right)^{\text{n}}}$$ for n = -1 and n = +1, respectively, are”. This question is standard in competitive examination syllabi for Differential Equations.

$${\text{x}} < 1,\,{\text{x}} > 1$$ is the correct answer to “The solution of $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = {{\text{y}}^2}$$ with initial value y(0) = 1 bounded in the interval”. This question is standard in competitive examination syllabi for Differential Equations.

$$\left[ {{{\text{c}}_1} + {{\text{c}}_2}{\text{x}} + {{\text{c}}_3}\sin \sqrt {3{\text{x}}} + {{\text{c}}_4}\cos \sqrt {3{\text{x}}} } \right]{\text{ and }}\left[ {3{{\text{x}}^4} - 12{{\text{x}}^2} + {\text{c}}} \right]$$ is the correct answer to “The respective expressions for complimentary function and particular integral part of the solution of the differential equation $$\frac{{{{\text{d}}^4}{\text{y}}}}{{{\text{d}}{{\text{x}}^4}}} + 3\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} = 108{{\text{x}}^2}$$ are”. This question is standard in competitive examination syllabi for Differential Equations.

(1 + 2t)e-t is the correct answer to “The solution of the differential equation $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{t}}^2}}} + 2\frac{{{\text{dy}}}}{{{\text{dt}}}} + {\text{y}} = 0$$ with y(0) = y'(0) = 1 is”. This question is standard in competitive examination syllabi for Differential Equations.

(-2, 2) is the correct answer to “Consider the differential equation $$\left( {{{\text{t}}^2} - 81} \right)\frac{{{\text{dy}}}}{{{\text{dt}}}} + 5{\text{ty}} = \sin \left( {\text{t}} \right)$$ with y(1) = 2π. There exists a unique solution for this differential equation when t belongs to the interval”. This question is standard in competitive examination syllabi for Differential Equations.

family of hyperbolas is the correct answer to “The solution to 6yy' - 25x = 0 represents a”. This question is standard in competitive examination syllabi for Differential Equations.

y = 1 - x + tan(x + c), where c is a constant is the correct answer to “Which one of the following is the general solution of the first order differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = {\left( {{\text{x}} + {\text{y}} - 1} \right)^2},$$ where x, y are real”. This question is standard in competitive examination syllabi for Differential Equations.

$${\text{y}} = \left( {{{\text{c}}_1} + {{\text{c}}_2}{\text{x}}} \right){{\text{e}}^{ - \frac{{{\text{ax}}}}{2}}}$$ is the correct answer to “If roots of the auxiliary equation of $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + {\text{a}}\frac{{{\text{dy}}}}{{{\text{dx}}}} + {\text{by}} = 0$$ are real and equal, the general solution of the differential equation is”. This question is standard in competitive examination syllabi for Differential Equations.

y = 36.95 e-2x is the correct answer to “The solution of the ordinary differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} + {\text{2y}} = 0$$ for the boundary condition, y = 5 at x = 1 is”. This question is standard in competitive examination syllabi for Differential Equations.

0 is the correct answer to “If y is the solution of the differential equation $$\eqalign{ & {{\text{y}}^3}\frac{{{\text{dy}}}}{{{\text{dx}}}} + {{\text{x}}^3} = 0, \cr & {\text{y}}\left( 0 \right) = 1 \cr} $$ the value of y(-1) is”. This question is standard in competitive examination syllabi for Differential Equations.

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

y = e-x - e2x + xe2x is the correct answer to “The solution of the differential equation, $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} - \frac{{{\text{dy}}}}{{{\text{dx}}}} - 2{\text{y}} = 3{{\text{e}}^{2{\text{x}}}},$$ where, y(0) = 0 and y'(0) = -2 is”. This question is standard in competitive examination syllabi for Differential Equations.

$$ - \frac{1}{3}{{\text{e}}^{ - 3{\text{x}}}} + {\text{K}}$$ is the correct answer to “With K as a constant, the solution possible for the first order differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = {{\text{e}}^{ - 3{\text{x}}}}$$ is”. This question is standard in competitive examination syllabi for Differential Equations.

x(t) = x0 e-3t is the correct answer to “The solution of the first order differential equation x'(t) = -3x(t), x(0) = x 0 is”. This question is standard in competitive examination syllabi for Differential Equations.

x-1 is the correct answer to “The solution of the equation $${\text{x}}\frac{{{\text{dy}}}}{{{\text{dx}}}} + {\text{y}} = 0$$ passing through the point (1, 1) is”. This question is standard in competitive examination syllabi for Differential Equations.

$${\text{x}} = \frac{{\text{t}}}{2}$$ is the correct answer to “With initial condition x(1) = 0.5, the solution of the differential equation, $${\text{t}}\frac{{{\text{dx}}}}{{{\text{dt}}}} + {\text{x}} = {\text{t}}$$ is”. This question is standard in competitive examination syllabi for Differential Equations.

x(t) = 2e-2t - e-4t is the correct answer to “For the differential equation $$\frac{{{{\text{d}}^2}{\text{x}}}}{{{\text{d}}{{\text{t}}^2}}} + 6\frac{{{\text{dx}}}}{{{\text{dt}}}} + 8{\text{x}} = 0$$ with initial conditions x(0) = 1 and $${\left. {\frac{{{\text{dx}}}}{{{\text{dy}}}}} \right|_{{\text{t}} = 0}} = 0,$$ the solution is”. This question is standard in competitive examination syllabi for Differential Equations.

$$\frac{{{\text{dv}}}}{{{\text{dt}}}} + \left( {1 - {\text{n}}} \right){\text{pv}} = \left( {1 - {\text{n}}} \right){\text{q}}$$ is the correct answer to “Transformation to linear form by substituting v = y 1 - n of the equation $$\frac{{{\text{dy}}}}{{{\text{dt}}}} + {\text{p}}\left( {\text{t}} \right){\text{y}} = {\text{q}}\left( {\text{t}} \right){{\text{y}}^{\text{n}}};\,{\text{n}} > 0$$ will be”. This question is standard in competitive examination syllabi for Differential Equations.

-e-2 is the correct answer to “Consider the differential equation \[{\rm{\ddot y}} + 2{\rm{\dot y}} + {\rm{y}} = 0\] with boundary conditions y(0) = 1, y(1) = 0. The value of y(2) is”. This question is standard in competitive examination syllabi for Differential Equations.

$$\frac{3}{7}{{\text{e}}^{ - \frac{7}{3}}}$$ is the correct answer to “For the equation, $$\frac{{{\text{dy}}}}{{{\text{dx}}}} + 7{{\text{x}}^2}{\text{y}} = 0,$$ if $${\text{y}}\left( {\text{0}} \right) = \frac{3}{7},$$ then the value of y(1) is”. This question is standard in competitive examination syllabi for Differential Equations.

1 is the correct answer to “Consider two solutions x(t) = x 1 (t) and x(t) = x 2 (t) of the differential equation $$\frac{{{{\text{d}}^2}{\text{x}}\left( {\text{t}} \right)}}{{{\text{d}}{{\text{t}}^2}}} + {\text{x}}\left( {\text{t}} \right) = 0,\,{\text{t}} > 0,$$ such that $${{\text{x}}_2} = 0,\,{\left. {\frac{{{\text{d}}{{\text{x}}_2}\left( {\text{t}} \right)}}{{{\text{dt}}}}} \right|_{{\text{t}} = 0}} = 1.$$ The Wronskian \[{\text{W}}\left( {\text{t}} \right) = \left| {\begin{array}{*{20}{c}} {{{\text{x}}_1}\left( {\text{t}} \right)}&{{{\text{x}}_2}\left( {\text{t}} \right)} \\ {\frac{{{\text{d}}{{\text{x}}_1}\left( {\text{t}} \right)}}{{{\text{dt}}}}}&{\frac{{{\text{d}}{{\text{x}}_2}\left( {\text{t}} \right)}}{{{\text{dt}}}}} \end{array}} \right|\] at $${\text{t}} = \frac{\pi }{2}$$ is”. This question is standard in competitive examination syllabi for Differential Equations.

cos t is the correct answer to “If $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{t}}^2}}} + {\text{y}} = 0$$ under the conditions y = 1, $$\frac{{{\text{dy}}}}{{{\text{dt}}}} = 0,$$ when t = 0, then y is equal to”. This question is standard in competitive examination syllabi for Differential Equations.

y = P cos x + Q sin x is the correct answer to “The general solution of $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} + {\text{y}} = 0$$ is”. This question is standard in competitive examination syllabi for Differential Equations.

$${\text{y}} = {{\text{e}}^{\frac{{{{\text{x}}^3}}}{3}}}$$ is the correct answer to “The solution for the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = {{\text{x}}^2}{\text{y}}$$ with the condition that y = 1 at x = 0 is”. This question is standard in competitive examination syllabi for Differential Equations.

y = cekx is the correct answer to “The solution of the differential equation $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = {\text{ky}},\,{\text{y}}\left( 0 \right) = {\text{c}}$$ is”. This question is standard in competitive examination syllabi for Differential Equations.

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

12 is the correct answer to “If y = 2x 3 - 3x 2 + 3x - 10, the value of ∆ 3 y will be (where, ∆ is forward differences operator)”. This question is standard in competitive examination syllabi for Differential Equations.

a-2, b-3, c-3, d-1 is the correct answer to “Match List-I with List-II and select the correct answer: List-I List-II A. $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \frac{{\text{y}}}{{\text{x}}}$$ 1. Circles B. $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = - \frac{{\text{y}}}{{\text{x}}}$$ 2. Straight lines C. $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = \frac{{\text{x}}}{{\text{y}}}$$ 3. Hyperbolas D. $$\frac{{{\text{dy}}}}{{{\text{dx}}}} = - \frac{{\text{x}}}{{\text{y}}}$$”. This question is standard in competitive examination syllabi for Differential Equations.

ae-t + bte-t is the correct answer to “If a and b are constants, the most general solution of the differential equation $$\frac{{{{\text{d}}^2}{\text{x}}}}{{{\text{d}}{{\text{t}}^2}}} + 2\frac{{{\text{dx}}}}{{{\text{dt}}}} + {\text{x}} = 0$$ is”. This question is standard in competitive examination syllabi for Differential Equations.

te-tu(t) is the correct answer to “The solution of the differential equation, for t > 0, y''(t) + 2y'(t) + y(t) = 0 with initial conditions y(0) = 0 and y'(0) = 1, is (u(t) denotes the unit step function),”. This question is standard in competitive examination syllabi for Differential Equations.

$${\text{y}} = {{\text{C}}_1}{{\text{x}}^2} + {{\text{C}}_2}{\text{x}} + 2$$ is the correct answer to “A differential equation is given as $${{\text{x}}^2}\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} - 2{\text{x}}\frac{{{\text{dy}}}}{{{\text{dx}}}} + 2{\text{y}} = 4$$ The solution of differential equation in terms of arbitrary constant C 1 and C 2 is”. This question is standard in competitive examination syllabi for Differential Equations.

y = x + c where C is an arbitrary constant is the correct answer to “The solution of the differential equation $$\frac{{{{\text{d}}^2}{\text{y}}}}{{{\text{d}}{{\text{x}}^2}}} = 0$$ with boundary conditions $${\text{i}}{\text{.}}\,\frac{{{\text{dy}}}}{{{\text{dx}}}} = 1{\text{ at x}} = 0;\,{\text{ii}}{\text{.}}\,\frac{{{\text{dy}}}}{{{\text{dx}}}} = 1{\text{ at x}} = 1{\text{ is}}$$”. This question is standard in competitive examination syllabi for Differential Equations.