What is the solution for y^{''}+16y=0.5sin(4t),y(0)=0.5 ?

The solution for y^{''}+16y=0.5sin(4t),y(0)=0.5 is y=0.5cos(4t)+c_{2}sin(4t)-0.0625tcos(4t)

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y^{''}+16y=0.5sin(4t),y(0)=0.5

y^{^{''}} + 16 y = 0.5 sin (4 t), y (0) = 0.5

y = 0.5 cos (4 t) + c_{2} sin (4 t) - 0.0625 t cos (4 t)

Solution steps

y^{''} (t) + 16 y = 0.5 sin (4 t)

Solve linear ODE:

y = 0.5 cos (4 t) + c_{2} sin (4 t) - 0.0625 t cos (4 t)

y = 0.5 cos (4 t) + c_{2} sin (4 t) - 0.0625 t cos (4 t)

What is the solution for y^{''}+16y=0.5sin(4t),y(0)=0.5 ?
The solution for y^{''}+16y=0.5sin(4t),y(0)=0.5 is y=0.5cos(4t)+c_{2}sin(4t)-0.0625tcos(4t)