What is the solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=4sin(x) ?

The solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=4sin(x) is y=c_{1}e^x+c_{2}xe^x+2cos(x)

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(d^2y)/(dx^2)-2(dy)/(dx)+y=4sin(x)

\frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} + y = 4 sin (x)

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

Solution steps

\frac{d ^{2} y}{d x ^{2}} - 2 \frac{d y}{d x} + y = 4 sin (x)

Solve linear ODE:

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

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

y^{^{''}} - 4 y^{^{'}} + 3 y = e^{x}

y^{^{''}} + 4 y^{^{'}} + 5 y = \frac{1}{x} e^{2 x} sin (x)

y^{^{''}} + 4 y = - 16 e^{- 2 x} + 2 sin (x)

y^{^{''}} + 4 y^{^{'}} = 8 x + 1

y^{^{''}} - y = 2 (e^{x} e^{- x})^{- 1}

What is the solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=4sin(x) ?
The solution for (d^2y)/(dx^2)-2(dy)/(dx)+y=4sin(x) is y=c_{1}e^x+c_{2}xe^x+2cos(x)