What is the solution for 5y^{''}+cy^'+45y=0,y(0)=-8,y^'(0)=0 ?

The solution for 5y^{''}+cy^'+45y=0,y(0)=-8,y^'(0)=0 is

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5y^{''}+cy^'+45y=0,y(0)=-8,y^'(0)=0

5 y^{^{''}} + c y^{^{'}} + 45 y = 0, y (0) = - 8, y^{^{'}} (0) = 0

y = e^{\frac{t ( - c + c ^{2} - 900 )}{10}} (- 8 + \frac{4 ( c ^{2} - 900 - c )}{c ^{2} - 900}) - \frac{4 e ^{\frac{t ( - c - c ^{2} - 900 )}{10}} ( c ^{2} - 900 - c )}{c ^{2} - 900}

Solution steps

5 y^{''} (t) + c y^{'} (t) + 45 y = 0

Solve linear ODE:

y = e^{\frac{t ( - c + c ^{2} - 900 )}{10}} (- 8 + \frac{4 ( c ^{2} - 900 - c )}{c ^{2} - 900}) - \frac{4 e ^{\frac{t ( - c - c ^{2} - 900 )}{10}} ( c ^{2} - 900 - c )}{c ^{2} - 900}

y = e^{\frac{t ( - c + c ^{2} - 900 )}{10}} (- 8 + \frac{4 ( c ^{2} - 900 - c )}{c ^{2} - 900}) - \frac{4 e ^{\frac{t ( - c - c ^{2} - 900 )}{10}} ( c ^{2} - 900 - c )}{c ^{2} - 900}

What is the solution for 5y^{''}+cy^'+45y=0,y(0)=-8,y^'(0)=0 ?
The solution for 5y^{''}+cy^'+45y=0,y(0)=-8,y^'(0)=0 is