Second Order Linear Nonhomogeneous Differential Equations

[Mikesgto]

Homework Statement

y''+7y'=392sin(7t)+686cos(7t) with y(0)=4 and y'(0)=9

Homework Equations

No real relevant equations

The Attempt at a Solution

I assumed since the g(t) has function of both sine and cosine the solution would be both the real and non real parts of the solution to

y''+7y'=1078e^(7it)

my solution came out to be
77/24tcos(7t)+539/24tsin(7t)+i[77/24tsin(7t)-539/24tcos(7t)]
but given the initial conditions, i don't know what happen to my C's.
Also, if anyone could explain this method in a way that is easy to understand, I would appreciate it. I tried following the class notes, but it was of little help.
Thank you.

 

[Mark44]

Homework Statement

y''+7y'=392sin(7t)+686cos(7t) with y(0)=4 and y'(0)=9

Homework Equations

No real relevant equations

The Attempt at a Solution

I assumed since the g(t) has function of both sine and cosine the solution would be both the real and non real parts of the solution to

y''+7y'=1078e^(7it)

my solution came out to be
77/24tcos(7t)+539/24tsin(7t)+i[77/24tsin(7t)-539/24tcos(7t)]
but given the initial conditions, i don't know what happen to my C's.
Also, if anyone could explain this method in a way that is easy to understand, I would appreciate it. I tried following the class notes, but it was of little help.
Thank you.

1. Find the solutions to the homogeneous problem y'' + 7y' = 0.
2. Find a particular solution to the nonhomogeneous problem y'' + 7y' = 392sin(7t)+686cos(7t). y_p = A sin(7t) + B cos(7t) is a good place to start.
3. Form the general solution to the nonhomogeneous equation. Use the initial conditions to find the coefficients of the functions in step 1.