See this thread.
Hint: treat each time interval separately, and match the solutions in the different intervals by requiring u and u' to be continuous functions of t.
I started this problem by trying to solve for the complimentary solution. The characteristic equation is r^2 + 1 =0. Meaning r = i.
Thus, I got:
Using the initial conditions, I got c1 and c2 =0. I don't think this is right though...
First solve . You should get . Putting u(0)= 0, u'(0)= 0 into that gives you and so while , The solution to the entire equation is for [tex]0\le t\le \pi[/itex]. Since and , at we have and .
Now solve with initial conditions and . That will give you u for . Determine and from that.
Finally, solve with initial conditions [tex]u(2\pi)[/math ] and equal to the values you just got.
The problem is very similar to this...
Indicating with the LT of the DE is written in terms of the complex variable as...
Since is the solution is the 'only' , and because is...
The solution is then...