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

The solution for y^{''}-4y^'+4y=e^{2x}*sin(x) is y=c_{1}e^{2x}+c_{2}xe^{2x}-e^{2x}sin(x)

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

y^{^{''}} - 4 y^{^{'}} + 4 y = e^{2 x} \cdot sin (x)

y = c_{1} e^{2 x} + c_{2} x e^{2 x} - e^{2 x} sin (x)

Solution steps

y^{''} (x) - 4 y^{'} (x) + 4 y = e^{2 x} \cdot sin (x)

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} x e^{2 x} - e^{2 x} sin (x)

y = c_{1} e^{2 x} + c_{2} x e^{2 x} - e^{2 x} sin (x)

\frac{d ^{2} x}{d t ^{2}} + \frac{d x}{d t} = 6 t

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

y^{^{''}} + 2 y^{^{'}} = 49 e^{x} sin (2 x)

y^{^{''}} + 2 y^{^{'}} + 1 = t e^{t} sin (t)

y^{^{''''}} + 2 y^{^{''}} + y = 3 t + 10

What is the solution for y^{''}-4y^'+4y=e^{2x}*sin(x) ?
The solution for y^{''}-4y^'+4y=e^{2x}*sin(x) is y=c_{1}e^{2x}+c_{2}xe^{2x}-e^{2x}sin(x)