Find a fundamental set of solutions {y_(1)(x) , y_(2)(x)} of the equation (d^(2)y)/(dx^(2)) + 16 (dy)/(dx) + 73y=0

Question

Find a fundamental set of solutions {
`y_(1)(x) , y_(2)(x)`} of the equation


`(d^(2)y)/(dx^(2)) + 16 (dy)/(dx) + 73y=0`

Answer 1

`(\mathrm d^2y)/(\mathrm dx^2)+16(\mathrm dy)/(\mathrm dx)+73y=0\implies r^2+16r+73=0`

`\implies r=-8\pm3i`

`\implies\begin{cases}y_1=e^(-8x)\cos3x\\y_2=e^(-8x)\sin3x\end{cases}`