What is the solution for y^{''}-5y^'+6y=-4e^{3x}+10e^x-12 ?

The solution for y^{''}-5y^'+6y=-4e^{3x}+10e^x-12 is y=c_{1}e^{3x}+c_{2}e^{2x}-4xe^{3x}+5e^x-2

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y^{''}-5y^'+6y=-4e^{3x}+10e^x-12

y^{^{''}} - 5 y^{^{'}} + 6 y = - 4 e^{3 x} + 10 e^{x} - 12

y = c_{1} e^{3 x} + c_{2} e^{2 x} - 4 x e^{3 x} + 5 e^{x} - 2

Solution steps

y^{''} (x) - 5 y^{'} (x) + 6 y = - 4 e^{3 x} + 10 e^{x} - 12

Solve linear ODE:

y = c_{1} e^{3 x} + c_{2} e^{2 x} - 4 x e^{3 x} + 5 e^{x} - 2

y = c_{1} e^{3 x} + c_{2} e^{2 x} - 4 x e^{3 x} + 5 e^{x} - 2

y^{^{''}} - y^{^{'}} = - 3 x + e^{x}

y^{^{''}} + y = cos (- t + \frac{π}{2})

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

y^{^{''}} + 3 y^{^{'}} + 2 y = e^{x}

x^{^{''}} + 4 x^{^{'}} + 13 x = 5

What is the solution for y^{''}-5y^'+6y=-4e^{3x}+10e^x-12 ?
The solution for y^{''}-5y^'+6y=-4e^{3x}+10e^x-12 is y=c_{1}e^{3x}+c_{2}e^{2x}-4xe^{3x}+5e^x-2