What is the solution for y^{''}-y^'-6y=5-6^x,y(0)=2,y^'(0)=4 ?

The solution for y^{''}-y^'-6y=5-6^x,y(0)=2,y^'(0)=4 is

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y^{''}-y^'-6y=5-6^x,y(0)=2,y^'(0)=4

y^{^{''}} - y^{^{'}} - 6 y = 5 - 6^{x}, y (0) = 2, y^{^{'}} (0) = 4

y = (- c_{2} + \frac{5}{2} - \frac{- 5 ln ^{2} ( 6 ) + 5 ln ( 6 ) + 15}{15 ln ^{2} ( 6 ) - 15 ln ( 6 ) - 90}) e^{3 x} + c_{2} e^{- 2 x} - \frac{1}{2} + \frac{( ln ( 6 ) + 2 ) e ^{3 x} ( - 5 ( ln ( 6 ) - 3 ) - 2 ^{x} \cdot 3 ^{x + 1} ) + ( ln ( 6 ) - 3 ) \cdot 3 ^{\frac{l n ( 6 ) x + 5 x + l n ( 6 )}{l n ( 6 )}} \cdot 2 ^{\frac{l n ( 6 ) x + 5 x}{l n ( 6 )}} e ^{- 2 x}}{5 ln ^{2} ( 6 ) \cdot 3 ^{\frac{3 x + l n ( 6 )}{l n ( 6 )}} \cdot 2 ^{\frac{3 x}{l n ( 6 )}} - 5 ln ( 6 ) \cdot 3 ^{\frac{3 x + l n ( 6 )}{l n ( 6 )}} \cdot 2 ^{\frac{3 x}{l n ( 6 )}} - 5 \cdot 3 ^{1 + \frac{3 x + l n ( 6 )}{l n ( 6 )}} \cdot 2 ^{1 + \frac{3 x}{l n ( 6 )}}}

Solution steps

y^{''} (x) - y^{'} (x) - 6 y = 5 - 6^{x}

Solve linear ODE:

y = (- c_{2} + \frac{5}{2} - \frac{- 5 ln ^{2} ( 6 ) + 5 ln ( 6 ) + 15}{15 ln ^{2} ( 6 ) - 15 ln ( 6 ) - 90}) e^{3 x} + c_{2} e^{- 2 x} - \frac{1}{2} + \frac{( ln ( 6 ) + 2 ) e ^{3 x} ( - 5 ( ln ( 6 ) - 3 ) - 2 ^{x} \cdot 3 ^{x + 1} ) + ( ln ( 6 ) - 3 ) \cdot 3 ^{\frac{l n ( 6 ) x + 5 x + l n ( 6 )}{l n ( 6 )}} \cdot 2 ^{\frac{l n ( 6 ) x + 5 x}{l n ( 6 )}} e ^{- 2 x}}{5 ln ^{2} ( 6 ) \cdot 3 ^{\frac{3 x + l n ( 6 )}{l n ( 6 )}} \cdot 2 ^{\frac{3 x}{l n ( 6 )}} - 5 ln ( 6 ) \cdot 3 ^{\frac{3 x + l n ( 6 )}{l n ( 6 )}} \cdot 2 ^{\frac{3 x}{l n ( 6 )}} - 5 \cdot 3 ^{1 + \frac{3 x + l n ( 6 )}{l n ( 6 )}} \cdot 2 ^{1 + \frac{3 x}{l n ( 6 )}}}

y = (- c_{2} + \frac{5}{2} - \frac{- 5 ln ^{2} ( 6 ) + 5 ln ( 6 ) + 15}{15 ln ^{2} ( 6 ) - 15 ln ( 6 ) - 90}) e^{3 x} + c_{2} e^{- 2 x} - \frac{1}{2} + \frac{( ln ( 6 ) + 2 ) e ^{3 x} ( - 5 ( ln ( 6 ) - 3 ) - 2 ^{x} \cdot 3 ^{x + 1} ) + ( ln ( 6 ) - 3 ) \cdot 3 ^{\frac{l n ( 6 ) x + 5 x + l n ( 6 )}{l n ( 6 )}} \cdot 2 ^{\frac{l n ( 6 ) x + 5 x}{l n ( 6 )}} e ^{- 2 x}}{5 ln ^{2} ( 6 ) \cdot 3 ^{\frac{3 x + l n ( 6 )}{l n ( 6 )}} \cdot 2 ^{\frac{3 x}{l n ( 6 )}} - 5 ln ( 6 ) \cdot 3 ^{\frac{3 x + l n ( 6 )}{l n ( 6 )}} \cdot 2 ^{\frac{3 x}{l n ( 6 )}} - 5 \cdot 3 ^{1 + \frac{3 x + l n ( 6 )}{l n ( 6 )}} \cdot 2 ^{1 + \frac{3 x}{l n ( 6 )}}}

y^{^{''}} - 2 y^{^{''}} + y = x e^{x} + e^{x}

x^{^{''}} + 16 x = 34 e^{- 2 t} cos (4 t)

x^{^{''}} + 9 x = \frac{sin ( t )}{9}

x^{^{''}} - 6 x + 9 x = 126 t^{2} e^{3 t}

y^{^{''}} + 9 \cdot y^{^{'}} = 5 t

What is the solution for y^{''}-y^'-6y=5-6^x,y(0)=2,y^'(0)=4 ?
The solution for y^{''}-y^'-6y=5-6^x,y(0)=2,y^'(0)=4 is