What is the solution for y^{''}+2ay^'+(a^2+1)y=0,y(0)=1,y^'(0)=0 ?

The solution for y^{''}+2ay^'+(a^2+1)y=0,y(0)=1,y^'(0)=0 is y=e^{-at}(cos(t)+asin(t))

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y^{''}+2ay^'+(a^2+1)y=0,y(0)=1,y^'(0)=0

y^{^{''}} + 2 a y^{^{'}} + (a^{2} + 1) y = 0, y (0) = 1, y^{^{'}} (0) = 0

y = e^{- a t} (cos (t) + a sin (t))

Solution steps

y^{''} (t) + 2 a y^{'} (t) + (a^{2} + 1) y = 0

Solve linear ODE:

y = e^{- a t} (cos (t) + a sin (t))

y = e^{- a t} (cos (t) + a sin (t))

What is the solution for y^{''}+2ay^'+(a^2+1)y=0,y(0)=1,y^'(0)=0 ?
The solution for y^{''}+2ay^'+(a^2+1)y=0,y(0)=1,y^'(0)=0 is y=e^{-at}(cos(t)+asin(t))