GATE CE
Engineering Mathematics
Differential Equations
Previous Years Questions

## Marks 1

The solution of the equation $$\,{{dQ} \over {dt}} + Q = 1$$ with $$Q=0$$ at $$t=0$$ is
Consider the following second $$-$$order differential equation : $$\,y''\,\, - 4y' + 3y = 2t - 3{t^2}\,\,\,$$ The particular solution of the differe...
Consider the following partial differential equation: $$\,\,3{{{\partial ^2}\phi } \over {\partial {x^2}}} + B{{{\partial ^2}\phi } \over {\partial x... The type of partial differential equation$${{{\partial ^2}p} \over {\partial {x^2}}} + {{{\partial ^2}p} \over {\partial {y^2}}} + 3{{{\partial ^2}p...
The integrating factor for the differential equation $${{dP} \over {dt}} + {k_2}\,P = {k_1}{L_0}{e^{ - {k_1}t}}\,\,$$ is
The solution of the differential equation $${{dy} \over {dx}} + {y \over x} = x$$ with the condition that $$y=1$$ at $$x=1$$ is
The order and degree of a differential equation $${{{d^3}y} \over {d{x^3}}} + 4\sqrt {{{\left( {{{dy} \over {dx}}} \right)}^3} + {y^2}} = 0$$ are res...
The partial differential equation that can be formed from $$z=ax+by+ab$$ has the form $$\,\,\left( {p = {{\partial z} \over {\partial x}},q = {{\part... Solution of the differential equation$$3y{{dy} \over {dx}} + 2x = 0$$represents a family of The degree of the differential equation$$\,{{{d^2}x} \over {d{t^2}}} + 2{x^3} = 0\,\,$$is A body originally at$${60^ \circ }$$cools down to$$40$$in$$15$$minutes when kept in air at a temperature of$${25^ \circ }$$c. What will be the ... The solution of the differential equation$$\,{x^2}{{dy} \over {dx}} + 2xy - x + 1 = 0\,\,\,$$given that at$$x=1,y=0$$is The number of boundary conditions required to solve the differential equation$$\,\,{{{\partial ^2}\phi } \over {\partial {x^2}}} + {{{\partial ^2}\p...
If $$c$$ is a constant, then the solution of $${{dy} \over {dx}} = 1 + {y^2}$$ is
For the differential equation $$f\left( {x,y} \right){{dy} \over {dx}} + g\left( {x,y} \right) = 0\,\,$$ to be exact is
The differential equation $${y^{11}} + {\left( {{x^3}\,\sin x} \right)^5}{y^1} + y = \cos {x^3}\,\,\,\,$$ is
The necessary & sufficient condition for the differential equation of the form $$\,\,M\left( {x,y} \right)dx + N\left( {x,y} \right)dy = 0\,\,$$ t...

## Marks 2

The solution of the partial differential equation $${{\partial u} \over {\partial t}} = \alpha {{{\partial ^2}u} \over {\partial {x^2}}}$$ is of the ...
Consider the following second order linear differential equation $${{{d^2}y} \over {d{x^2}}} = - 12{x^2} + 24x - 20$$ The boundary conditions are: a...
Consider the following differential equation $$x\left( {y\,dx + x\,dy} \right)\cos \left( {{y \over x}} \right)$$ $$\,\,\,\,\,\,\,\,\,\, = y\left( ... Water is following at a steady rate through a homogeneous and saturated horizontal soil strip of$$10$$m length. The strip is being subjected to a con... The solution of the ordinary differential equation$${{dy} \over {dx}} + 2y = 0$$for the boundary condition,$$y=5$$at$$x=1$$is The solution to the ordinary differential equation$${{{d^2}y} \over {d{x^2}}} + {{dy} \over {dx}} - 6y = 0\,\,\,$$is The solution for the differential equation$$\,{{d\,y} \over {d\,x}} = {x^2}\,y$$with the condition that$$y=1$$at$$x=0$$is Transformation to linear form by substituting$$v = {y^{1 - n}}$$of the equation$${{dy} \over {dt}} + p\left( t \right)y = q\left( t \right){y^n},\...
The solution $${{{d^2}y} \over {d{x^2}}} + 2{{dy} \over {dx}} + 17y = 0;$$ $$y\left( 0 \right) = 1,{\left( {{{d\,y} \over {d\,x}}} \right)_{x = {\rai... Biotransformation of an organic compound having concentration$$(x)$$can be modeled using an ordinary differential equation$$\,{{d\,x} \over {dt}} ...
The solution for the following differential equation with boundary conditions $$y(0)=2$$ and $$\,\,{y^1}\left( 1 \right) = - 3$$ is where $${{{d^2... Solve$${{{d^4}y} \over {d{x^4}}} - y = 15\,\cos \,\,2x$$The differential equation$${{dy} \over {dx}} + py = Q,$$is a linear equation of first order only if, Solve$${{{d^4}v} \over {d{x^4}}} + 4{\lambda ^4}v = 1 + x + {x^2}$$The solution of a differential equation$${y^{11}} + 3{y^1} + 2y = 0$$is of the form The differential equation$${{{d^4}y} \over {d{x^4}}} + P{{{d^2}y} \over {d{x^2}}} + ky = 0\,\, is
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