Differential Equations · Engineering Mathematics · GATE IN

The figure shows the plot of $$y$$ as a function of $$x$$

The function shown in the solution of the differential equation (assuming all initial conditions to be zero) is

The type of the partial differential equation $${{\partial f} \over {\partial t}} = {{{\partial ^2}f} \over {\partial {x^2}}}\,\,is$$

With initial condition $$x\left( 1 \right)\,\,\, = \,\,\,\,0.5,\,\,\,$$ the solution of the differential equation, $$\,\,\,t{{dx} \over {dt}} + x = t\,\,\,$$ is

Consider the differential equation $${{dy} \over {dx}} + y = {e^x}$$ with $$y(0)=1.$$ Then the value of $$y(1)$$ is

Consider the differential equation $${{dy} \over {dx}} = 1 + {y^2}.$$ Which one of the following can be particular solution of this differential equation ?

The general solution of the differential equation $$\left( {{D^2} - 4D + 4} \right)y = 0$$ is of the form (given $$D = {d \over {dx}}$$ and $${C_1},{C_2}$$ are constants)

The maximum value of the solution $$y$$ $$(t)$$ of the differential equation $$\,\,y\left( t \right) + \mathop y\limits^{ \bullet \,\, \bullet } \left( t \right) = 0\,\,\,$$ with initial conditions $$\,\,\mathop y\limits^ \bullet \left( 0 \right) = 1\,\,$$ and $$\,\,y\left( 0 \right) = 1,\,\,$$ for $$\,t \ge 0\,\,$$ is

Consider the differential equation $$\mathop y\limits^{ \bullet \bullet } + 2\,\mathop y\limits^ \bullet + y = 0\,\,$$ with boundary conditions $$y(0)=1$$ & $$y(1)=0.$$ The value of $$y(2)$$ is

For initial value problem $$\,\mathop y\limits^{ \bullet \bullet } + 2\,\mathop y\limits^ \bullet + \left( {101} \right)y = \left( {10.4} \right){e^x},y\left( 0 \right) = 1.1\,\,$$ and $$y(0)=-0.9.$$ Various solutions are written in the following groups. Match the type of solution with the correct expression.

Group-$${\rm I}$$
$$P.$$$$\,\,\,\,$$ General solution of Homogeneous equations
$$Q.$$$$\,\,\,\,$$ Particular integral
$$R.$$$$\,\,\,\,$$ Total solution satisfying boundary Conditions

Group-$${\rm II}$$
$$(1)$$$$\,\,\,\,$$ $$0.1\,{e^x}$$
$$(2)$$$$\,\,\,\,$$ $$\,{e^{ - x}}\left[ {A\,\cos \,10x + B\,\sin \,10x} \right]$$
$$(3)$$$$\,\,\,\,$$ $${e^{ - x}}\,\cos \,10x + 0.1\,{e^x}$$