What is the solution for 64y^{''}+80y^'+25y=0,y(0)=a,y^'(0)=-5 ?

The solution for 64y^{''}+80y^'+25y=0,y(0)=a,y^'(0)=-5 is y=ae^{-(5t)/8}+(-5+(5a)/8)e^{-(5t)/8}t

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64y^{''}+80y^'+25y=0,y(0)=a,y^'(0)=-5

64 y^{^{''}} + 80 y^{^{'}} + 25 y = 0, y (0) = a, y^{^{'}} (0) = - 5

y = a e^{- \frac{5 t}{8}} + (- 5 + \frac{5 a}{8}) e^{- \frac{5 t}{8}} t

Solution steps

64 y^{''} (t) + 80 y^{'} (t) + 25 y = 0

Solve linear ODE:

y = a e^{- \frac{5 t}{8}} + (- 5 + \frac{5 a}{8}) e^{- \frac{5 t}{8}} t

y = a e^{- \frac{5 t}{8}} + (- 5 + \frac{5 a}{8}) e^{- \frac{5 t}{8}} t

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What is the solution for 64y^{''}+80y^'+25y=0,y(0)=a,y^'(0)=-5 ?
The solution for 64y^{''}+80y^'+25y=0,y(0)=a,y^'(0)=-5 is y=ae^{-(5t)/8}+(-5+(5a)/8)e^{-(5t)/8}t