- #1
EvilKermit
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Homework Statement
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 infinitely many solutions?
d) For what values of A do there exist no solutions?
Homework Equations
y'' + y = A + sin(2x)
y(0) = y'([tex]\pi[/tex]/2) = 2
The Attempt at a Solution
y = yh + yp
0 = 1+ [tex]\lambda[/tex]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 infinite 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?