The differential equation is a second-order linear homogeneous differential equation with constant coefficients. It can be solved using the method of characteristic equations.
The characteristic equation is found by setting the characteristic polynomial equal to zero:
r^2 + 3r + 2 = 0
Solving this equation, we get: r = -1, -2
These are the roots of the characteristic equation, which are real and distinct.
Therefore, the general solution to the differential equation is:
f(x) = c1e^(-x) + c2e^(-2x) + Asin(x) + Bcos(x)
Where c1, c2, A, B are arbitrary constants determined by initial or boundary conditions.
We can use the initial conditions f(0) = 1 and f'(0) = 2, to find the values of c1, c2, A, B.
f(0) = 1 = c1e^(0) + c2e^(0) + A = c1 + c2 + A
f'(0) = 2 = -c1e^(0) - 2c2e^(0) + B = -c1 - 2c2 + B
Therefore, c1 = 1 - A - c2 and c2 = (2 - B)/2
Substituting the value of c1 and c2 in the general solution, we get:
f(x) = (e^(-x)(1 - A - c2) + e^(-2x)(2 - B)/2) + Asin(x) + Bcos(x)
This is the solution to the differential equation f''(x) + 3f'(x) + 2f(x) = sin(x) with the initial conditions f(0) = 1, f'(0) = 2.
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