What is the solution for y^{''}+10/3 y^'+ay=0,y(0)=1,y^'(0)= 5/3 ?

The solution for y^{''}+10/3 y^'+ay=0,y(0)=1,y^'(0)= 5/3 is

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y^{''}+10/3 y^'+ay=0,y(0)=1,y^'(0)= 5/3

y^{^{''}} + \frac{10}{3} y^{^{'}} + a y = 0, y (0) = 1, y^{^{'}} (0) = \frac{5}{3}

y = e^{\frac{t ( - 5 + - 9 a + 25 )}{3}} (1 + \frac{- - 9 a + 25 + 10}{2 25 - 9 a}) - \frac{e ^{- \frac{t ( - 9 a + 25 + 5 )}{3}} ( - - 9 a + 25 + 10 )}{2 25 - 9 a}

Solution steps

y^{''} (t) + \frac{10}{3} y^{'} (t) + a y (t) = 0

Solve linear ODE:

y = e^{\frac{t ( - 5 + - 9 a + 25 )}{3}} (1 + \frac{- - 9 a + 25 + 10}{2 25 - 9 a}) - \frac{e ^{- \frac{t ( - 9 a + 25 + 5 )}{3}} ( - - 9 a + 25 + 10 )}{2 25 - 9 a}

y = e^{\frac{t ( - 5 + - 9 a + 25 )}{3}} (1 + \frac{- - 9 a + 25 + 10}{2 25 - 9 a}) - \frac{e ^{- \frac{t ( - 9 a + 25 + 5 )}{3}} ( - - 9 a + 25 + 10 )}{2 25 - 9 a}

u^{^{''}} + u^{^{'}} = 0

9 y^{^{''}} - 30 y^{^{'}} - 24 y = 0, y (0) = 0, y^{^{'}} (0) = 1

y^{^{''}} + 5 y^{^{'}} + 7 y = 0

\frac{d ^{2} y}{d x ^{2}} - 7 \frac{d y}{d x} + 6 y = 0

x^{^{''}} + 10 x^{^{'}} + 15 x = 0

What is the solution for y^{''}+10/3 y^'+ay=0,y(0)=1,y^'(0)= 5/3 ?
The solution for y^{''}+10/3 y^'+ay=0,y(0)=1,y^'(0)= 5/3 is