The one dimensional heat conduction partial difference equation $$\,\,{{\partial T} \over {\partial t}} = {{{\partial ^2}T} \over {\partial {x^2}}}\,\,$$ is

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 ratio $${y \over x}$$ (or) $${x \over y}$$

For the differential equation $$\,\,{{dy} \over {dt}} + 5y = 0\,\,$$ with $$y(0)=1,$$ the general solution is