Question: Given that \({{y}_{1}}\left( x \right)={{e}^{2x}}\cos x\) and \({{y}_{2}}\left( x \right)={{e}^{2x}}\sin x\) are solutions to the homogeneous equation

find the solutions to this equation that satisfy the following initial conditions:

(a) \(y\left( 0 \right)=2,\,\,y'\left( 0 \right)=1\)

(b) \(y\left( \pi \right)=4{{e}^{2\pi }},\,\,\,y'\left( \pi \right)=5{{e}^{2\pi }}\)