Finding the particular solution to an ODE with set boundary conditions.

[Jack_O]

Homework Statement

Homework Equations

N/A

The Attempt at a Solution

The problem and attempt are as above, I'm not sure where to go from here though. I'm not sure what to do with the boundary condition of dx/dt=-2 and t=0.
Any help appreciated.

 

[djeitnstine]

The answer to your auxiliary equation is wrong, m=+/- 2i not +/- 2 so the homogeneous solution is of the form $$x_{h}=C_{1}cos(2t)+C_{2}Sin(2t)$$

 

[Jack_O]

Thanks for pointing that out, I've had another go and realized i needed to differentiate the general solution and then sub in the other boundaries to get the other simultaneous equation. My completed solution looks like this:

I have also done part c), got the answer x=(i/sqrt3)[-e^(2it)+e^(-2it)]

 

[djeitnstine]

make your solution without complex coefficients so $$y_{h}=C_{1}sin2t+C_{2}cos2t$$

then work out the result $$C_{1}sin0+C_{2}cos0=1$$
$$C_{2}=1$$

$$y_{h}'=2C_{1}cos2t-2C_{2}sin2t$$

$$2C_{1}cos0-2C_{2}0=-2$$

$$-2C_{1}-2=-2$$

$$C_{1}=0$$...

 

[Jack_O]

If i use Euler's formula to sub out e^(2it) for cos(2t)+i*sin(2t) won't i still be left with complex coefficients? Their are complex numbers in the (1+i) term. I don't see how i can cancel out all the complex numbers.

 

[djeitnstine]

I really don't want to go through the proof of why this is true so I'll point you here: http://tutorial.math.lamar.edu/Classes/DE/DE.aspx

basically C1 and C2 can be complex or real and can simply be left as constants.