Homework Help: Boundary value problem

1. Dec 8, 2009

EvilKermit

1. The problem statement, all variables and given/known data

a) Solve for the BVP. Where A is a real number.
b) For what values of A does there exist a unique solutions? What is the solution?
c) For what values of A do there exist infintely many solutions?
d) For what values of A do there exist no solutions?

2. Relevant equations

y'' + y = A + sin(2x)
y(0) = y'($$\pi$$/2) = 2

3. The attempt at a solution

y = yh + yp
0 = 1+ $$\lambda$$2
yh =c1*cos(x) + c2*cos(x)

yp = A + B*sin(2x)
y = A + B*sin(2x)
y'' = -4B*sin(2x)
A + sin(2x) = A -3B*sin(2x)
A = A, B = -1/3
yp = A - 1/3*sin(2x)

y = A - 1/3*sin(2x) + c1*cos(x) + c2*sin(x)
y' = - 2/3*sin(2x) - c1*sin(x) + c2*cos(x)
2 = A + c1
2 = 2/3 + c2
c2 = 8/3

y = A - 1/3*sin(2x) + c1*cos(x) + 8/3*sin(x)

I'm confused about answering the questions. A would be equal to all real numbers, since one could solve for c1. How can I give the solution? There is a unique solution for each value of A, which I would have to write infinte solutions. And there is no value of A when there is an inifiite amount of solutions or no values.

IF A is defined, what would the answer be?

2. Dec 9, 2009

HallsofIvy

So c1= 2-A

y= A- 1/2 sin(2x)+ (2- A)cos(x)+ 8/3 sin(x)

3. Dec 9, 2009

EvilKermit

So:
y= A- 1/2 sin(2x)+ (2- A)cos(x)+ 8/3 sin(x)

c) For what values of A do there exist infintely many solutions?
d) For what values of A do there exist no solutions?

Would c and d then be no values of A have infintely many solutions or nonexistent solution?

4. Dec 10, 2009

Yes.