MATH300 Midterm: MATH 300 UofA Exam Solution 5
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Solve the initial value problem y + 9y = f (t) y(0) = 0 y (0) = 0 where f (t) is the 2 -periodic input function given by its fourier series f (t) = Xn=1(cid:20) cos nt n (cid:21) . n2 + ( 1)n sin nt. The general solution to the homogeneous equation is therefore yh(t) = c1 cos 3t + c2 sin 3t where c1 and c2 are arbitrary constants. N = ( 1)n n(9 n2) for n 1, n 6= 3. While for n = 3, the term in the driving force has the same frequency as the natural frequency of the system, and we have to solve the nonhomogeneous equation. In this case the method of undetermined coe cients suggests a solution of the form y . 3 (t) + 9y3(t) = a3 cos 3t + b3 sin 3t. y3(t) = t( 3 cos 3t + 3 sin 3t).
