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Engineering Mathematics
Differential Equations
Previous Years Questions

## Marks 1

The differential equation $$\,{{{d^2}y} \over {d{x^2}}} + 16y = 0$$ for $$y(x)$$ with the two boundary conditions $${\left. {{{dy} \over {dx}}} \right... Consider the following partial differential equation for$$u(x,y)$$with the constant$$c>1:\,{{\partial u} \over {\partial y}} + c{{\partial ...
Consider the following differential equation $${{dy} \over {dt}} = - 5y;$$ initial condition: $$y=2$$ at $$t=0.$$ The value of $$y$$ at $$t=3$$ is ...
Find the solution of $${{{d^2}y} \over {d{x^2}}} = y$$ which passes through origin and the point $$\left( {ln2,{3 \over 4}} \right)$$
The solution of the initial value problem $$\,\,{{dy} \over {dx}} = - 2xy;y\left( 0 \right) = 2\,\,\,$$ is
The partial differential equation $$\,\,{{\partial u} \over {\partial t}} + u{{\partial u} \over {\partial x}} = {{{\partial ^2}u} \over {\partial {x^... Consider the differential equation$${{dy} \over {dx}} = \left( {1 + {y^2}} \right)x\,\,.$$The general solution with constant$$'C'$$is The blasius equation$$\,{{{d^3}f} \over {d{\eta ^3}}} + {f \over 2}\,{{{d^2}f} \over {d{\eta ^2}}} = 0\,\,\,\,$$is a The solution of$$x{{dy} \over {dx}} + y = {x^4}$$with condition$$y\left( 1 \right) = {6 \over 5}$$Given that$$\mathop x\limits^{ \bullet \bullet } + 3x = 0$$and$$x\left( 0 \right) = 1,\,\,\mathop x\limits^ \bullet \left( 0 \right) = 1,$$Wha... The solution of$${{d\,y} \over {d\,x}} = {y^2}$$with initial value$$y(0)=1$$is bounded in the interval is The partial differential equation$$\,\,{{{\partial ^2}\phi } \over {\partial {x^2}}} + {{{\partial ^2}\phi } \over {\partial {y^2}}} + {{\partial \ph...
For $$\,\,\,{{{d^2}y} \over {d{x^2}}} + 4{{dy} \over {dx}} + 3y = 3{e^{2x}},\,\,$$ the particular integral is
The solution of the differential equation $${{dy} \over {dx}} + 2xy = {e^{ - {x^2}}}\,\,$$ with $$y(0)=1$$ is
If $${x^2}\left( {{{d\,y} \over {d\,x}}} \right) + 2xy = {{2\ln x} \over x}$$ and $$y(1)=0$$ then what is $$y(e)$$?
The solution of the differential equation $${{dy} \over {dx}} + {y^2} = 0$$ is
The equation $$\,\,\,{{{d^2}u} \over {d{x^2}}} + \left( {{x^2} + 4x} \right){{dy} \over {dx}} + y = {x^8} - 8\,\,{u \over {{x^2}}} = 8.\,\,\,$$ is a
The general solution of the differential equation $${x^2}{{{d^2}y} \over {d{x^2}}} - x{{dy} \over {dx}} + y = 0$$ is
The particular solution for the differential equation $${{{d^2}y} \over {d{t^2}}} + 3{{dy} \over {dx}} + 2y = 5\cos x$$ is
The one dimensional heat conduction partial difference equation $$\,\,{{\partial T} \over {\partial t}} = {{{\partial ^2}T} \over {\partial {x^2}}}\,\... The solution to the differential equation$$\,{f^{11}}\left( x \right) + 4{f^1}\left( x \right) + 4f\left( x \right) = 0$$A differential equation of the form$${{dy} \over {dx}} = f\left( {x,y} \right)\,\,$$is homogeneous if the function$$f(x,y)$$depends only on the ... For the differential equation$$\,\,{{dy} \over {dt}} + 5y = 0\,\,$$with$$y(0)=1,$$the general solution is The differential equation$${y^{11}} + y = 0\,$$is subjected to the conditions$$y(0) = 0,\,\,\,y\left( \lambda \right) = 0.\,\,$$In order th... The differential$$\,\,\,{{{d^2}y} \over {d{x^2}}} + {{dy} \over {dx}} + \sin y = 0\,\,$$is ## Marks 2 Consider the differential equation$$\,\,3y''\left( x \right) + 27y\left( x \right) = 0\,\,$$with initial conditions$$y\left( 0 \right) = 0$$and$$...
If $$y = f(x)$$ satiesfies the boundary value problem $$\,\,y''\,\,\, + \,\,\,9y\,\,\, = \,\,\,0,\,\,\,y\left( 0 \right)\,\,\, = \,\,\,0,\,$$ $$\,\,y\... The general solution of the differential equation$$\,\,{{dy} \over {dx}} = \cos \left( {x + y} \right),\,\,$$with$$c$$as a constant, is Consider two solutions$$\,x\left( t \right)\,\,\,\, = \,\,\,{x_1}\left( t \right)\,\,$$and$$x\left( t \right)\,\,\,\, = \,\,\,{x_2}\left( t \rig...
The matrix form of the linear system $${{dx} \over {dt}} = 3x - 5y$$ and $$\,{{dy} \over {dt}} = 4x + 8y\,\,$$ is
If $$\,y = f\left( x \right)\,\,$$ is the solution of $${{{d^2}y} \over {d{x^2}}} = 0$$ with the boundary conditions $$y=5$$ at $$x=0,$$ and $$\,{{d... The solution to the differential equation$$\,{{{d^2}u} \over {d{x^2}}} - k{{du} \over {dx}} = 0\,\,\,$$where$$'k'$$is a constant, subjected to th... Consider the differential equation$$\,\,{x^2}{{{d^2}y} \over {d{x^2}}} + x{{dy} \over {dx}} - 4y = 0\,\,\,$$with the boundary conditions of$$\,\,y\...
It is given that $$y'' + 2y' + y = 0,\,\,\,\,y\left( 0 \right) = 0,y\left( 1 \right) = 0.$$ What is $$y(0.5)$$?
The complete solution of the ordinary differential equation $${{{d^2}y} \over {d\,{x^2}}} + p{{dy} \over {dx}} + qy = 0$$ is $$\,y = {c_1}\,{e^{ - x}... Which of the following is a solution of the differential equation$$\,{{{d^2}y} \over {d{x^2}}} + p{{dy} \over {dx}} + \left( {q + 1} \right)y = 0?$$... Find the solution of the differential equation$$\,{{{d^2}u} \over {d{t^2}}} + {\lambda ^2}y = \cos \left( {wt + k} \right)$$with initial conditio... The radial displacement in a rotating disc is governed by the differential equation$$\,\,{{{d^2}u} \over {d{x^2}}} + {1 \over x}{{du} \over {dx}} - {...
Solve for $$y$$ if $${{{d^2}y} \over {d{t^2}}} + 2{{dy} \over {dt}} + y = 0$$ with $$y(0)=1$$ and $${y^1}\left( 0 \right) = 2$$
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