What is the solution for y^{''}-6y^'+8y=5x^2sin(4x) ?

The solution for y^{''}-6y^'+8y=5x^2sin(4x) is

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y^{''}-6y^'+8y=5x^2sin(4x)

y^{^{''}} - 6 y^{^{'}} + 8 y = 5 x^{2} sin (4 x)

y = c_{1} e^{4 x} + c_{2} e^{2 x} - \frac{x ^{2} sin ( 4 x )}{16} + \frac{3 x ^{2} cos ( 4 x )}{16} - \frac{3 x sin ( 4 x )}{20} + \frac{7 x cos ( 4 x )}{160} - \frac{227}{6400} sin (4 x) - \frac{189}{6400} cos (4 x)

Solution steps

y^{''} (x) - 6 y^{'} (x) + 8 y = 5 x^{2} sin (4 x)

Solve linear ODE:

y = c_{1} e^{4 x} + c_{2} e^{2 x} - \frac{x ^{2} sin ( 4 x )}{16} + \frac{3 x ^{2} cos ( 4 x )}{16} - \frac{3 x sin ( 4 x )}{20} + \frac{7 x cos ( 4 x )}{160} - \frac{227}{6400} sin (4 x) - \frac{189}{6400} cos (4 x)

y = c_{1} e^{4 x} + c_{2} e^{2 x} - \frac{x ^{2} sin ( 4 x )}{16} + \frac{3 x ^{2} cos ( 4 x )}{16} - \frac{3 x sin ( 4 x )}{20} + \frac{7 x cos ( 4 x )}{160} - \frac{227}{6400} sin (4 x) - \frac{189}{6400} cos (4 x)

1 x^{^{''}} - 3 x^{^{'}} + 2 x = 20 sin (2 t)

y^{^{''}} + 4 y = 0.5

y^{^{'''}} + 2 y^{^{'}} = cos (x)

y^{^{'''}} = 3 y^{^{''}} + 9 (e^{2 x}) + 5 cos (x)

(D^{4} - 1) y = e^{- x}

What is the solution for y^{''}-6y^'+8y=5x^2sin(4x) ?
The solution for y^{''}-6y^'+8y=5x^2sin(4x) is