# Problem 3. Complex exponential functions are commonly used in electrical engineering to model sinusoidal inputs to circuits. In particular, if one wants to solve a problem involving a differential equation of the form ay" + by' + cy = Rsin wt or ay" + by' + cy Rcos wt, it is actually easier to solve the equation ay" + by' + cy = Rewt (1) and then use Euler's identity to isolate the solutions to the two original equations. The reason that (1) is easier to handle is that complex exponential functions behave just like we would expect when taking derivatives: d (eiwt) = iweiwt dt This fact allows us to apply the Method of Undetermined Coefficients to find a particular solution yp(t) to (1) in the usual way (albeit with some complex numbers floating around). Our particular solution will have the form Yp(t) = f(t) + ig(t), where f and g are real-valued functions. Since Euler's identity tells us that Reut iwt = R cos wt +iRsin wt it turns out that the real part f(t) is a particular solution to the equation ay" + by' + cy = Rcos wt, while the imaginary part g(t) is a particular solution to ay" + by' + cy = Rsin wt. Use this approach to find particular solutions to the following differential equations. (a) y" – 2y' + y = 4 sin t (b) y" + 16y = 3 cos 4t (Hint: Think about the homogeneous solution first.)

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