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