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.)

Question
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
Expand
Transcribed Image Text

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.)
Expand
Transcribed Image Text

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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