GATE-CH-1994-1-c-math-1mark

Integrating factor for the differential equation ${\displaystyle \frac{dy}{dx}}+P(x)y=Q(x)$ is

• $e^{\int Pdx}$

• $e^{-\int Pdx}$

• $\int Pdx$

• $dP/dx$

GATE-CH-1994-1-e-math-1mark

The solution for the differential equation ${\displaystyle \frac{d^{2}y}{d{x}^2}}+5{\displaystyle \frac{dy}{dx}}+6y=0$ is

• $C_1{e}^{-2x}+C_2{e}^{3x}$

• $C_1\sin 2x+C_2\cos 2x$

• $C_1{e}^{2x}+C_2{e}^{-3x}$

• $C_1{e}^{-2x}+C_2{e}^{-3x}$

GATE-CH-2000-1-3-math-1mark

The integrating factor for the differential equation: $({\cos }^{2}x){\displaystyle \frac{dy}{dx}}+y=\tan x$, is

• $e^{\tan x}$

• $\cos 2x$

• $e^{-\tan x}$

• $\sin 2x$

GATE-CH-2004-4-math-1mark

The differential equation ${\displaystyle \frac{d^{2}y}{d{x}^2}+\sin x\frac{dy}{dx}+y{e}^x=\sinh x}$ is

• first order and linear

• first order and non-linear

• second order and linear

• second order and non-linear

GATE-CH-2005-1-math-1mark

Match the following, where $x$ is the spatial coordinate and $t$ is time.

Group I	Group II
P) Wave equation	I) ${\displaystyle \frac{\partial c}{\partial t}=\alpha \frac{\partial c}{\partial x}}$
Q) Heat equation	II) ${\displaystyle \frac{\partial c}{\partial t}={\alpha }^2\frac{{\partial }^{2}c}{\partial x^2}}$
	III) ${\displaystyle \frac{{\partial }^{2}c}{\partial t^2}={\alpha }^2\frac{\partial c}{\partial x}}$
	IV) ${\displaystyle \frac{{\partial }^{2}c}{\partial t^2}={\alpha }^2\frac{{\partial }^{2}c}{\partial x^2}}$

• P - IV, Q - II

• P - II, Q - IV

• P - III, Q - I

• P - I, Q - III

GATE-CH-2008-2-math-1mark

Which ONE of the following is NOT a solution of the differential equation ${\displaystyle \frac{d^{2}y}{d{x}^2}+y=1}$? ___________

• $y=1$

• $y=1+\cos x$

• $y=1+\sin x$

• $y=2+\sin x+\cos x$

GATE-CH-2012-2-math-1mark

If $a$ and $b$ are arbitrary constants, then the solution to the differential equation ${\displaystyle \frac{d^{2}y}{d{x}^2}-4y=0}$ is

• $y=ax+b$

• $y=a{e}^{-x}$

• $y=a\sin 2x+b\cos 2x$

• $y=a\cosh 2x+b\sinh 2x$

GATE-CH-1995-3-a-math-2mark

Match the items in the left column with the appropriate items in the right column.

• I. $y=x^2$

Answer 1A. linear O.D.E.B. nonlinear O.D.E.D. nonlinear algebraic equationC. linear algebraic equation

• II. $dy/dx=2x$

Answer 2A. linear O.D.E.B. nonlinear O.D.E.D. nonlinear algebraic equationC. linear algebraic equation

GATE-CH-1998-2-2-math-2mark

The differential equation ${\displaystyle \frac{d^{2}x}{d{t}^2}}+3{\displaystyle \frac{dx}{dt}}+2x=0$ will have a solution of the form

• $C_1{e}^{3t}+C_2{e}^{2t}$

• $C_1{e}^{-2t}+C_2{e}^{-t}$

• $C_1{e}^{-3t}+C_2{e}^{-2t}$

• $C_1{e}^{-5t}$

GATE-CH-2000-2-4-math-2mark

The general solution of ${\displaystyle \frac{d^{4}y}{d{x}^4}}+2{\displaystyle \frac{d^{2}y}{d{x}^2}}+y=0$ is ___________ (where $C_1,C_2,C_3$, and $C_4$ are constants).

• $(C_{1}x+C_2)e^x+(C_3+C_{4}x)e^{-x}$

• $C_1\cos x+C_2\sin x+C_3{e}^x+C_4{e}^{-x}$

• $C_1{e}^{ix}+C_2{e}^{-ix}$

• $(C_1+C_{2}x)\cos x+(C_3+C_{4}x)\sin x$

GATE-CH-2003-32-math-2mark

The value of $y$ as $t\to \mathrm{\infty }$ for the following differential equation for an initial value of $y(1)=0$ is $$(4t^2+1)\frac{dy}{dt}+8yt-t=0$$

• 1

• 1/2

• 1/4

• 1/8

GATE-CH-2003-36-math-2mark

The differential equation ${\displaystyle \frac{d^{2}x}{d{t}^2}+10\frac{dx}{dt}+25x=0}$ will have a solution of the form ___________ (where $C_1$ and $C_2$ are constants).

• $(C_1+C_{2}t)e^{-5t}$

• $C_1{e}^{-2t}$

• $C_1{e}^{-5t}+C_2{e}^{5t}$

• $C_1{e}^{-5t}+C_2{e}^{2t}$

GATE-CH-2004-35-math-2mark

The differential equation for the variation of the amount of salt $x$ in a tank with time $t$ is given by ${\displaystyle \frac{dx}{dt}+\frac{x}{20}=10}$. $x$ is in kg and $t$ is in minutes. Assuming that there is no salt in the tank initially, the time (in min) at which the amount of salt increases to 100 kg is

• $10\ln 2$

• $20\ln 2$

• $50\ln 2$

• $100\ln 2$

GATE-CH-2005-36-math-2mark

What condition is to be satisfied so that the solution of the differential equation $$\frac{d^{2}y}{d{x}^2}+a\frac{dy}{dx}+by=0$$ is of the form $y=(C_1+C_{2}x)e^{mx}$, where $C_1$ and $C_2$ are constants of integration?

• $a^2=b$

• $b^2=a$

• $a^2=4b$

• $b^2=4a$

GATE-CH-2008-21-math-2mark

Which ONE of the following transformations $\{u=f(y)\}$ reduces $$\frac{dy}{dx}+A{y}^3+By=0$$ to a linear differential equation? ($A$ and $B$ are positive constants)

• $u=y^{-3}$

• $u=y^{-2}$

• $u=y^{-1}$

• $u=y^2$

GATE-CH-2009-22-math-2mark

The general solution of the differential equation $$\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}-6y=0$$ with $C_1$ and $C_2$ as constants of integration, is

• $C_1{e}^{-3x}+C_2{e}^{-2x}$

• $C_1{e}^{3x}+C_2{e}^{-2x}$

• $C_1{e}^{-3x}+C_2{e}^{2x}$

• $C_1{e}^{-3x}+C_2{e}^{2x}$

GATE-CH-2010-26-math-2mark

The solution of the differential equation $$\frac{d^{2}y}{d{t}^2}+2\frac{dy}{dt}+2y=0$$ with the initial conditions ${\displaystyle y(0)=0,{\frac{dy}{dt}|}_{t=0}=-1}$, is

• $-t\sin t$

• $-e^{-t}(1-\cos t)$

• $-(t+\sin t)/2$

• $-e^{-t}\sin t$

GATE-CH-2011-27-math-2mark

Which one of the following choices is a solution of the differential equation given below? $$\frac{dy}{dx}=\frac{y^2}{x}+\frac{y}{x}-\frac{2}{x}$$ Note: $c$ is a real constant.

• ${\displaystyle y=\frac{c-x^2}{c+2x^2}}$

• ${\displaystyle y=\frac{c+2x^2}{c-x^2}}$

• ${\displaystyle y=\frac{c-x^3}{c+2x^3}}$

• ${\displaystyle y=\frac{c+2x^3}{c-x^3}}$

GATE-CH-2013-27-math-2mark

The solution of the differential equation ${\displaystyle \frac{dy}{dx}-y^2=0,}$ given $y=1$ at $x=0$ is

• ${\displaystyle \frac{1}{1+x}}$

• ${\displaystyle \frac{1}{1-x}}$

• ${\displaystyle \frac{1}{(1-x{)}^2}}$

• ${\displaystyle \frac{x^3}{3}+1}$

GATE-CH-2013-28-math-2mark

The solution of the differential equation ${\displaystyle \frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}+0.25y=0}$, given $y=0$ at $x=0$ and ${\displaystyle \frac{dy}{dx}=1}$ at $x=0$ is

• $x{e}^{0.5x}-x{e}^{-0.5x}$

• $0.5x{e}^x-0.5x{e}^{-x}$

• $x{e}^{0.5x}$

• $-x{e}^{0.5x}$

GATE-CH-2014-27-math-2mark

The integrating factor for the differential equation ${\displaystyle \frac{dy}{dx}-\frac{y}{1+x}=(1+x)}$ is

• ${\displaystyle \frac{1}{1+x}}$

• $(1+x)$

• $x(1+x)$

• ${\displaystyle \frac{x}{1+x}}$

GATE-CH-2014-28-math-2mark

The differential equation ${\displaystyle \frac{d^{2}y}{d{x}^2}+x^2\frac{dy}{dx}+x^{3}y=e^x}$ is a

• non-linear differential equation of first degree

• linear differential equation of first degree

• linear differential equation of second degree

• non-linear differential equation of second degree

GATE-CH-2016-27-math-2mark

What is the solution for the second order differential equation ${\displaystyle \frac{d^{2}y}{d{x}^2}+y=0}$, with the initial conditions $y{|}_{x=0}=5$ and ${\displaystyle {\frac{dy}{dx}|}_{x=0}=10}$ ?

• $y=5+10\sin x$

• $y=5\cos x-5\sin x$

• $y=5\cos x+10x$

• $y=5\cos x+10\sin x$

GATE-EC-2017-S1-29-math-2mark

Which one of the following is the general solution of the first order differential equation $$\frac{dy}{dx}=(x+y-1{)}^2$$ where $x,y$ are real?

• $y=1+x+{\tan }^{-1}(x+c)$ where $c$ is a constant

• $y=1+x+\tan (x+c)$ where $c$ is a constant

• $y=1-x+{\tan }^{-1}(x+c)$ where $c$ is a constant

• $y=1-x+\tan (x+c)$ where $c$ is a constant

GATE-CH-1994-2-c-math-1mark

$Mdx+Ndy$ is an exact differential when ---------

• Answer

GATE-CH-1994-2-g-math-1mark

The differential equation ${\displaystyle \frac{d^{2}y}{d{x}^2}}+y=0$, with the conditions $y(0)=0$ and $y(1)=1$ is called a --------- value problem.

• Answer

GATE-CH-1995-3-b-math-2mark

Match the items in the left column with the appropriate items in the right column.

• Answer

GATE-CH-1996-10-math-5mark

Solve $${\displaystyle \frac{dy}{dx}}+0.6y=6e^{-0.5x}$$ using the integrating factor method, given $y=1$ at $x=0$.

• Answer

GATE-CH-1999-4-math-5mark

Solve ${\displaystyle \frac{dy}{dx}}-6xy=-6x$ by the following methods:

1. variation of parameters
2. separation of variables

• Answer