What is the solution for y^{''}-8y^'+16y=e^{4x}*ln(9+x^2) ?

The solution for y^{''}-8y^'+16y=e^{4x}*ln(9+x^2) is

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y^{''}-8y^'+16y=e^{4x}*ln(9+x^2)

y^{^{''}} - 8 y^{^{'}} + 16 y = e^{4 x} \cdot ln (9 + x^{2})

y = c_{1} e^{4 x} + c_{2} x e^{4 x} + \frac{1}{2} e^{4 x} x^{2} ln (x^{2} + 9) - \frac{3}{2} e^{4 x} x^{2} + 6 e^{4 x} x arctan (\frac{x}{3}) - \frac{9}{2} e^{4 x} ln (x^{2} + 9) + \frac{9}{2} e^{4 x}

Solution steps

y^{''} (x) - 8 y^{'} (x) + 16 y = e^{4 x} \cdot ln (9 + x^{2})

Solve linear ODE:

y = c_{1} e^{4 x} + c_{2} x e^{4 x} + \frac{1}{2} e^{4 x} x^{2} ln (x^{2} + 9) - \frac{3}{2} e^{4 x} x^{2} + 6 e^{4 x} x arctan (\frac{x}{3}) - \frac{9}{2} e^{4 x} ln (x^{2} + 9) + \frac{9}{2} e^{4 x}

y = c_{1} e^{4 x} + c_{2} x e^{4 x} + \frac{1}{2} e^{4 x} x^{2} ln (x^{2} + 9) - \frac{3}{2} e^{4 x} x^{2} + 6 e^{4 x} x arctan (\frac{x}{3}) - \frac{9}{2} e^{4 x} ln (x^{2} + 9) + \frac{9}{2} e^{4 x}

y^{^{''}} - 4 y = (x - 2) e^{x}

y^{^{''}} + 4 y^{^{'}} + 3 y = 15

(D^{2} + 1) y = sec^{3} (x)

y^{^{''}} - 4 y^{^{'}} + 3 y = x, y (0) = 6, y^{^{'}} (0) = 10

y^{^{''}} + 6 y^{^{'}} + 25 y = x^{2}

What is the solution for y^{''}-8y^'+16y=e^{4x}*ln(9+x^2) ?
The solution for y^{''}-8y^'+16y=e^{4x}*ln(9+x^2) is