Two solutions to y′′−8y′+32y=0\displaystyle {y}{''}-{8}{y}'+{32}{y}={0} are y1=e4tsin⁡(4t)\displaystyle {y}_{{1}}={e}^{{{4}{t}}}{\sin{{\left({4}{t}\right)}}}, y2=e4tcos⁡(4t)\displaystyle {y}_{{2}}={e}^{{{4}{t}}}{\cos{{\left({4}{t}\right)}}}.

a) Find the Wronskian.

W\displaystyle {W} =  

b) Find the solution satisfying the initial conditions y(0)=−2, y′(0)=−24\displaystyle {y}{\left({0}\right)}=-{2},\ {y}'{\left({0}\right)}=-{24}

y=\displaystyle {y}=  

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