What is the solution for y^{''}+16y=e^{5t},y(0)=0,y^'(0)=0 ?

The solution for y^{''}+16y=e^{5t},y(0)=0,y^'(0)=0 is y=-1/41 cos(4t)-5/164 sin(4t)+1/41 e^{5t}

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y^{''}+16y=e^{5t},y(0)=0,y^'(0)=0

y^{^{''}} + 16 y = e^{5 t}, y (0) = 0, y^{^{'}} (0) = 0

y = - \frac{1}{41} cos (4 t) - \frac{5}{164} sin (4 t) + \frac{1}{41} e^{5 t}

Solution steps

y^{''} (t) + 16 y = e^{5 t}

Solve linear ODE:

y = - \frac{1}{41} cos (4 t) - \frac{5}{164} sin (4 t) + \frac{1}{41} e^{5 t}

y = - \frac{1}{41} cos (4 t) - \frac{5}{164} sin (4 t) + \frac{1}{41} e^{5 t}

y^{^{''}} - 4 y^{^{'}} + 4 y = e^{2 x} + x^{2}

y^{^{'''}} + 10 y^{^{''}} + 25 y^{^{'}} = e x

y^{^{''}} - y^{^{'}} + 2 y = x

2 y^{^{''}} + 2 y^{^{'}} + 2 y = sin (t) + 1

u^{^{''}} + a u^{^{'}} (t) = b

What is the solution for y^{''}+16y=e^{5t},y(0)=0,y^'(0)=0 ?
The solution for y^{''}+16y=e^{5t},y(0)=0,y^'(0)=0 is y=-1/41 cos(4t)-5/164 sin(4t)+1/41 e^{5t}