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

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

• \(e^{\int Pdx}\)

• \(e^{-\int Pdx}\)

• \(\int Pdx\)

• \(dP/dx\)

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

The solution for the differential equation \(\dfrac {d^2y}{dx^2} + 5\dfrac {dy}{dx} + 6y = 0\) 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: \((\cos ^2x)\dfrac {dy}{dx} + y = \tan x\), is

• \(e^{\tan x}\)

• \(\cos 2x\)

• \(e^{-\tan x}\)

• \(\sin 2x\)

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

The differential equation \(\displaystyle \frac {d^2y}{dx^2} + \sin x \frac {dy}{dx} + ye^x = \sinh x \) 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 \(\displaystyle \frac {d^2y}{dx^2} + y = 1\)? ___________

• \(y=1\)

• \(y=1+\cos x\)

• \(y=1+\sin x\)

• \(y=2+\sin x + \cos x\)

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

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

• \(y=ax+b\)

• \(y=ae^{-x}\)

• \(y=a\sin 2x + b\cos 2x\)

• \(y=a\cosh 2x + b\sinh 2x\)

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 \(\dfrac {d^2x}{dt^2} + 3\dfrac {dx}{dt} + 2x = 0\) 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 \(\dfrac {d^4y}{dx^4} + 2\dfrac {d^2y}{dx^2} + y = 0\) is ___________ (where \(C_1, C_2, C_3\), and \(C_4\) are constants).

• \((C_1 x + C_2)e^x + (C_3 + C_4x)e^{-x}\)

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

• \(C_1e^{ix} + C_2e^{-ix}\)

• \((C_1 + C_2x)\cos x + (C_3+C_4x)\sin x\)

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

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

• \((C_1 + C_2t)e^{-5t}\)

• \(C_1 e^{-2t}\)

• \(C_1 e^{-5t} + C_2 e^{5t}\)

• \(C_1 e^{-5t} + C_2 e^{2t}\)

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 \[ \frac {d^2y}{dx^2} + a\frac {dy}{dx} + by = 0\] is of the form \(y=(C_1+C_2x)e^{mx}\), 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 \[ \frac {dy}{dx} + Ay^3+By=0 \] 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 \[ \frac {d^2y}{dx^2} - \frac {dy}{dx} - 6y = 0 \] with \(C_1\) and \(C_2\) as constants of integration, is

• \(C_1e^{-3x}+C_2e^{-2x}\)

• \(C_1e^{3x}+C_2e^{-2x}\)

• \(C_1e^{-3x}+C_2e^{2x}\)

• \(C_1e^{-3x}+C_2e^{2x}\)

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

The solution of the differential equation \[ \frac {d^2y}{dt^2} + 2\frac {dy}{dt} + 2y = 0 \] with the initial conditions \(\displaystyle y(0)=0, \ \left .\frac {dy}{dt}\right |_{t=0} = -1\), 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 \(\ \displaystyle \frac {d^2y}{dx^2} - \frac {dy}{dx} + 0.25y =0\), given \(y=0\) at \(x=0\) and \(\displaystyle \ \frac {dy}{dx}=1\) at \(x=0\) is

• \(xe^{0.5x}-xe^{-0.5x}\)

• \(0.5xe^{x}-0.5xe^{-x}\)

• \(xe^{0.5x}\)

• \(-xe^{0.5x}\)

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 \(\ \displaystyle \frac {d^2y}{dx^2}+x^2\frac {dy}{dx}+x^3y=e^x \ \) 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 \(\displaystyle \frac {d^2y}{dx^2}+y=0\), with the initial conditions \(y|_{x=0}=5\) and \(\displaystyle \left .\frac {dy}{dx}\right |_{x=0}=10\) ?

• \(y=5+10\sin x\)

• \(y = 5\cos x - 5\sin x\)

• \(y = 5\cos x + 10x\)

• \(y = 5\cos x + 10\sin x\)

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

\(M dx + N dy\) is an exact differential when ---------

• Answer

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

The differential equation \(\dfrac {d^2y}{dx^2} + y = 0\), 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 \[ \dfrac {dy}{dx} + 0.6 y = 6e^{-0.5x} \] using the integrating factor method, given \(y=1\) at \(x=0\).

• Answer

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

Solve \(\dfrac {dy}{dx} - 6xy = -6x\) by the following methods:

1. variation of parameters
2. separation of variables

• Answer