GATE ME

Differential Equations

Engineering Mathematics

(Past Years Questions)

Marks 1

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The differential equation $$\,{{{d^2}y} \over {d{x^2}}} + 16y = 0$$ for $$y(x)$$ with the two boundary conditions $${\le...
GATE ME 2017 Set 1
Consider the following partial differential equation for $$u(x,y)$$ with the constant $$c>1:$$ $$\,{{\partial u} \over {...
GATE ME 2017 Set 1
Consider the following differential equation $${{dy} \over {dt}} = - 5y;$$ initial condition: $$y=2$$ at $$t=0.$$ The ...
GATE ME 2015 Set 2
Find the solution of $${{{d^2}y} \over {d{x^2}}} = y$$ which passes through origin and the point $$\left( {ln2,{3 \over...
GATE ME 2015 Set 1
The solution of the initial value problem $$\,\,{{dy} \over {dx}} = - 2xy;y\left( 0 \right) = 2\,\,\,$$ is
GATE ME 2014 Set 4
The partial differential equation $$\,\,{{\partial u} \over {\partial t}} + u{{\partial u} \over {\partial x}} = {{{\par...
GATE ME 2013
Consider the differential equation $${{dy} \over {dx}} = \left( {1 + {y^2}} \right)x\,\,.$$ The general solution with c...
GATE ME 2011
The blasius equation $$\,{{{d^3}f} \over {d{\eta ^3}}} + {f \over 2}\,{{{d^2}f} \over {d{\eta ^2}}} = 0\,\,\,\,$$ is a
GATE ME 2010
The solution of $$x{{dy} \over {dx}} + y = {x^4}$$ with condition $$y\left( 1 \right) = {6 \over 5}$$
GATE ME 2009
Given that $$\mathop x\limits^{ \bullet \bullet } + 3x = 0$$ and $$x\left( 0 \right) = 1,\,\,\mathop x\limits^ \bulle...
GATE ME 2008
The solution of $${{d\,y} \over {d\,x}} = {y^2}$$ with initial value $$y(0)=1$$ is bounded in the interval is
GATE ME 2007
The partial differential equation $$\,\,{{{\partial ^2}\phi } \over {\partial {x^2}}} + {{{\partial ^2}\phi } \over {\pa...
GATE ME 2007
The solution of the differential equation $${{dy} \over {dx}} + 2xy = {e^{ - {x^2}}}\,\,$$ with $$y(0)=1$$ is
GATE ME 2006
For $$\,\,\,{{{d^2}y} \over {d{x^2}}} + 4{{dy} \over {dx}} + 3y = 3{e^{2x}},\,\,$$ the particular integral is
GATE ME 2006
If $${x^2}\left( {{{d\,y} \over {d\,x}}} \right) + 2xy = {{2\ln x} \over x}$$ and $$y(1)=0$$ then what is $$y(e)$$?
GATE ME 2005
The solution of the differential equation $${{dy} \over {dx}} + {y^2} = 0$$ is
GATE ME 2003
The equation $$\,\,\,{{{d^2}u} \over {d{x^2}}} + \left( {{x^2} + 4x} \right){{dy} \over {dx}} + y = {x^8} - 8\,\,{u \ove...
GATE ME 1999
The general solution of the differential equation $${x^2}{{{d^2}y} \over {d{x^2}}} - x{{dy} \over {dx}} + y = 0$$ is
GATE ME 1998
The particular solution for the differential equation $${{{d^2}y} \over {d{t^2}}} + 3{{dy} \over {dx}} + 2y = 5\cos x$$ ...
GATE ME 1996
The one dimensional heat conduction partial difference equation $$\,\,{{\partial T} \over {\partial t}} = {{{\partial ^2...
GATE ME 1996
The solution to the differential equation $$\,{f^{11}}\left( x \right) + 4{f^1}\left( x \right) + 4f\left( x \right) = 0...
GATE ME 1995
A differential equation of the form $${{dy} \over {dx}} = f\left( {x,y} \right)\,\,$$ is homogeneous if the function $...
GATE ME 1995
For the differential equation $$\,\,{{dy} \over {dt}} + 5y = 0\,\,$$ with $$y(0)=1,$$ the general solution is
GATE ME 1994
The differential equation $${y^{11}} + y = 0\,$$ is subjected to the conditions $$y(0) = 0,$$ $$\,\,\,y\left( \lambda \...
GATE ME 1993
The differential $$\,\,\,{{{d^2}y} \over {d{x^2}}} + {{dy} \over {dx}} + \sin y = 0\,\,$$ is
GATE ME 1993

Marks 2

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Consider the differential equation $$\,\,3y''\left( x \right) + 27y\left( x \right) = 0\,\,$$ with initial conditions $$...
GATE ME 2017 Set 2
If $$y = f(x)$$ satiesfies the boundary value problem $$\,\,y''\,\,\, + \,\,\,9y\,\,\, = \,\,\,0,\,\,\,y\left( 0 \right)...
GATE ME 2016 Set 1
Consider two solutions $$\,x\left( t \right)\,\,\,\, = \,\,\,{x_1}\left( t \right)\,\,$$ and $$x\left( t \right)\,\,\...
GATE ME 2014 Set 3
The general solution of the differential equation $$\,\,{{dy} \over {dx}} = \cos \left( {x + y} \right),\,\,$$ with $$c$...
GATE ME 2014 Set 2
The matrix form of the linear system $${{dx} \over {dt}} = 3x - 5y$$ and $$\,{{dy} \over {dt}} = 4x + 8y\,\,$$ is
GATE ME 2014 Set 1
If $$\,y = f\left( x \right)\,\,$$ is the solution of $${{{d^2}y} \over {d{x^2}}} = 0$$ with the boundary conditions $...
GATE ME 2014 Set 1
The solution to the differential equation $$\,{{{d^2}u} \over {d{x^2}}} - k{{du} \over {dx}} = 0\,\,\,$$ where $$'k'$$ ...
GATE ME 2013
Consider the differential equation $$\,\,{x^2}{{{d^2}y} \over {d{x^2}}} + x{{dy} \over {dx}} - 4y = 0\,\,\,$$ with the b...
GATE ME 2012
It is given that $$y'' + 2y' + y = 0,\,\,\,\,y\left( 0 \right) = 0,y\left( 1 \right) = 0.$$ What is $$y(0.5)$$?
GATE ME 2008
The complete solution of the ordinary differential equation $${{{d^2}y} \over {d\,{x^2}}} + p{{dy} \over {dx}} + qy = 0$...
GATE ME 2005
Which of the following is a solution of the differential equation $$\,{{{d^2}y} \over {d{x^2}}} + p{{dy} \over {dx}} + \...
GATE ME 2005
Find the solution of the differential equation $$\,{{{d^2}u} \over {d{t^2}}} + {\lambda ^2}y = \cos \left( {wt + k} \r...
GATE ME 2000
The radial displacement in a rotating disc is governed by the differential equation $$\,\,{{{d^2}u} \over {d{x^2}}} + {1...
GATE ME 1998
Solve for $$y$$ if $${{{d^2}y} \over {d{t^2}}} + 2{{dy} \over {dt}} + y = 0$$ with $$y(0)=1$$ and $${y^1}\left( 0 \right...
GATE ME 1994

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