GATE ECE
Engineering Mathematics
Differential Equations
Previous Years Questions

## Marks 1

Consider the following partial differential equation (PDE) $$a{{{\partial ^2}f(x,y)} \over {\partial {x^2}}} + b{{{\partial ^2}f(x,y)} \over {\partial... The general solution of the differential equation$$\,\,{{{d^2}y} \over {d{x^2}}} + 2{{dy} \over {dx}} - 5y = 0\,\,\,$$in terms of arbitrary constant... Consider the following statements about the linear dependence of the real valued functions$${y_1} = 1,\,\,{y_2} = x$$and$${y_3} = {x^2}$$. Over the... The general solution of the differential equation$$\,\,{{dy} \over {dx}} = {{1 + \cos 2y} \over {1 - \cos 2x}}\,\,$$is Consider the differential equation$$\,\,{{dx} \over {dt}} = 10 - 0.2\,x$$with initial condition$$x(0)=1.$$The response$$x(t)$$for$$t > 0$$... If$$a$$and$$b$$are constants, the most general solution of the differential equation$$\,{{{d^2}x} \over {d{t^2}}} + 2{{dx} \over {dt}} + x = 0$$... If the characteristic equation of the differential equation$$\,{{{d^2}y} \over {d{x^2}}} + 2\alpha {{dy} \over {dx}} + y = 0\,\,$$has two equal root... With initial condition$$x\left( 1 \right)\,\,\, = \,\,\,\,0.5,\,\,\,$$the solution of the differential equation,$$\,\,\,t{{dx} \over {dt}} + x = t\...
The solution of differential equation $${{dy} \over {dx}} = ky,y\left( 0 \right) = C$$ is
The order of differential equation $$\,\,{{{d^2}y} \over {d{t^2}}} + {\left( {{{dy} \over {dx}}} \right)^3} + {y^4} = {e^{ - t}}\,\,$$ is
A solution of the differential equation $${{{d^2}y} \over {d{x^2}}} - 5{{dy} \over {dx}} + 6y = 0\,$$ is given by
The following differential equation has $$3{{{d^2}y} \over {d{t^2}}} + 4{\left( {{{dy} \over {dt}}} \right)^3} + {y^2} + 2 = x$$
$$y = {e^{ - 2x}}$$ is a solution of the differential equation $$\,{y^{11}} + {y^1} - 2y = 0$$

## Marks 2

The position of a particle y(t) is described by the differential equation : $${{{d^2}y} \over {d{t^2}}} = - {{dy} \over {dt}} - {{5y} \over 4}$$. The...
A curve passes through the point ($$x$$ = 1, $$y$$ = 0) and satisfies the differential equation $${{dy} \over {dx}} = {{{x^2} + {y^2}} \over {2y}} + {... Which one of the following is the general solution of the first order differential equation$${{dy} \over {dx}} = {\left( {x + y - 1} \right)^2}$$, ... The particular solution of the initial value problem given below is$$\,\,{{{d^2}y} \over {d{x^2}}} + 12{{dy} \over {dx}} + 36y = 0\,\,$$with$$\,y\l...
Consider the differential equation $${{{d^2}x\left( t \right)} \over {d{t^2}}} + 3{{dx\left( t \right)} \over {dt}} + 2x\left( t \right) = 0$$ Given...
The Solution of the differential equation $$\,{{{d^2}y} \over {d{t^2}}} + 2{{dy} \over {dt}} + y = 0\,\,$$ with $$\,y\left( 0 \right) = {y^1}\left( 0 ... With initial values$$\,\,\,y\left( 0 \right) = y'\left( 0 \right) = 1,\,\,\,$$the solution of the differential equation$$\,\,{{{d^2}y} \over {d{x^...
Which ONE of the following is a linear non - homogeneous differential equation , where $$x$$ and $$y$$ are the independent and dependent variables res...
A function $$n(x)$$ satisfies the differential equation $${{{d^2}n\left( x \right)} \over {d{x^2}}} - {{n\left( x \right)} \over {{L^2}}} = 0$$ where...
Match each differential equation in Group $$I$$ to its family of solution curves from Group $$II.$$ Group $$I$$ $$P:$$$$\,\,\,$$ $${{dy} \over {dx}... Which of the following is a solution to the differential equation$${d \over {dt}}x\left( t \right) + 3x\left( t \right) = 0,\,\,x\left( 0 \right) = 2...
The solution of the differential equation $${k^2}{{{d^2}y} \over {d\,{x^2}}} = y - {y_2}\,\,$$ under the boundary conditions (i) $$y = {y_1}$$ at $$x... For the differential equation$${{{d^2}y} \over {d{x^2}}} + {k^2}y = 0,$$the boundary conditions are (i)$$y=0$$for$$x=0$$and (ii)$$y=0$$fo... Solve the differential equation$${{{d^2}y} \over {d{x^2}}} + y = x\,\,$$with the following conditions$$(i)$$at$$x=0, y=1(ii)$$at$$x=0, $$... Match each of the items$$A, B, C$$with an appropriate item from$$1, 2, 3, 4$$and$$5$$List-$${\rm I}(P){a_1}{{{d^2}y} \over {d{x^2}}} +...
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