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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} + 2 y^{^{'}} + 6 y = e^{- 2 t}, y (0) = 2, y^{^{'}} (0) = 1

y^{^{''}} + 4 y^{^{'}} + 5 y = 1, y (0) = 0, y^{^{'}} (0) = 0

10 y^{^{''}} + 45 y^{^{'}} + 2500 y = 180 cos (7 π x)

y^{^{''}} + 4 y = sin (t), y (0) = 1, y^{^{'}} (0) = - 1

y^{^{''}} - 4 y = 5 e^{2 x} + 6, y^{^{'}} (0) = - 1, y (0) = 0

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