(Complex function method) Let L be a linear constant- coefficient differential operator, and consider the equation
L[x] = F0 cos Ωt,
According to the method of undetermined coefficients, we can find a particular solution xp(t) by seeking xp(t) = A cos Ωt + B sin Ωt (or, in exceptional cases, t to an integer power times that). A slightly simpler line of approach that is sometimes used is as follows. Consider, in place of (12.1),
L[w] = F0eiΩt,
Equation (12.2) is simpler than (12.1) in that to find a particular solution we need only one term, wp(t) = AeiΩt. (If 2 = a + ib is any complex number, it is standard to call Re z = a and Im z = b the real part and the imaginary part of z, respectively.) Because, according to Euler’s formula, eiΩt = cos Ωt + I sin Ωt, it follows that Re eiΩt = cos Ωt and eiΩt = sin Ωt . Since the forcing function in (12.1) is the real part of the forcing function in (12.2), it seems plausible that xp(t) should be the real part of wp(t). Thus, we have the following method: to find a particular solution to (12.1) consider instead the simpler equation (12.2). Solve for wp(t) by seeking wp(t) = AeiΩt, and then recover the desired xp(t) from xp(t) = Re wp(t). Use the method to obtain a particular solution to the given equation:
mx″ + cx′ + kx = F0 cos Ωt
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