What is the solution for y^{''}+by^'+64y=0,y(0)=2,y^'(0)=0 ?

The solution for y^{''}+by^'+64y=0,y(0)=2,y^'(0)=0 is

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y^{''}+by^'+64y=0,y(0)=2,y^'(0)=0

y^{^{''}} + b y^{^{'}} + 64 y = 0, y (0) = 2, y^{^{'}} (0) = 0

y = e^{\frac{t ( - b + b ^{2} - 256 )}{2}} (2 + \frac{b - b ^{2} - 256}{b ^{2} - 256}) - \frac{e ^{\frac{t ( - b - b ^{2} - 256 )}{2}} ( b - b ^{2} - 256 )}{b ^{2} - 256}

Solution steps

y^{''} (t) + b y^{'} (t) + 64 y = 0

Solve linear ODE:

y = e^{\frac{t ( - b + b ^{2} - 256 )}{2}} (2 + \frac{b - b ^{2} - 256}{b ^{2} - 256}) - \frac{e ^{\frac{t ( - b - b ^{2} - 256 )}{2}} ( b - b ^{2} - 256 )}{b ^{2} - 256}

y = e^{\frac{t ( - b + b ^{2} - 256 )}{2}} (2 + \frac{b - b ^{2} - 256}{b ^{2} - 256}) - \frac{e ^{\frac{t ( - b - b ^{2} - 256 )}{2}} ( b - b ^{2} - 256 )}{b ^{2} - 256}

y^{^{''}} - 10 y^{^{'}} + α y = 0

y^{^{'}} + y^{^{''}} = 0

y^{^{''}} + 9 y = 0, y (0) = 1, y^{^{'}} (0) = - 5

y^{^{''''}} - 40 y^{^{'''}} + 400 y^{^{''}} = 0

\frac{d ^{2} y}{d x ^{2}} + 11 \frac{d y}{d x} + 24 y = 0

What is the solution for y^{''}+by^'+64y=0,y(0)=2,y^'(0)=0 ?
The solution for y^{''}+by^'+64y=0,y(0)=2,y^'(0)=0 is