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Problem. Solve (D^{2} - 2D)y = e^{x}sin x

Solution. The complementary function is

y = c_{1} + c_{2}e^{2x}

We first form the relation

1) y = L_{1} + L_{2}e^{2x}

Differentiating 1) we get

We now set our first condition:

Equation 2) then becomes

Taking the derivative of 4) we get

We now set the last condition

We now solve equations 3) and 6) for and giving us

Integrating 7) and 8) gives

A particular solution of the given equation is then

and the general solution is

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