# Solving Linear Homogeneous Differential Equations

Consider the differential equation y''-2y'+2y = cost

(a) Find a general solution to this equation using techniques for solving linear homogeneous differential equations with constant coefficients in combination with a variation of parameters.

(b) Use Laplace Transforms to solve this differential equation with the initial conditions

y(0) = 1, y'(0) = 0

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Consider the differential equation y''-2y'+2y = cost

(a) Find a general solution to this equation using techniques for solving linear homogeneous differential equations with constant coefficients in combination with a variation of parameters.

Notice that the associated homogenous equation is linear second order with constant coefficients.

First, we "guess" a homogenous solution in the form:

Plugging this back into the equation we obtain:

The roots of the last equation are:

So the solution is:

As expected, we get two ...

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