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bhavik22
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Homework Statement
Use a Green's function to solve:
u" + 2u' + u = e-x
with u(0) = 0 and u(1) = 1 on 0[tex]\leq[/tex]x[tex]\leq[/tex]1
Homework Equations
This from the lecture notes in my course:
The Attempt at a Solution
Solving for the homogeneous equation first:
u" + 2u' + u = 0
From the characteristic equation,
[tex]\lambda[/tex]2 + 2[tex]\lambda[/tex] + 1 = 0
[tex]\lambda[/tex] = -1 (repeated root)
Characteristic solution:
u1(x) = c1e-x and u2 = c2e-xx
To satisfy boundary conditions,
u(0) = c1e0 + c2e0(0)
c1 = 0
Thus take f(x) = e-xx
and
u(1) = c1e-1 + c2e-1(1)
get c1 = [tex]\frac{e}{2}[/tex] and c2 = [tex]\frac{e}{2}[/tex]
Thus take g(x) = [tex]\frac{1}{2}[/tex]e-x+1(x+1)
evaluating the wronskian, W = [tex]\frac{1}{2}[/tex]e-2x+1
I contruct green's function as per the formula provided above and carry out the integral and get final answer of,
u(x) = xe-x(x2 - 3)
which obviously doesn't satisfy the second boundary condition of u(1) = 1.
I found out the final solution from Wolfram alpha:
u = [tex]\frac{1}{2}[/tex]e-xx(x+2e-1)
I've also tried many different combinations of f(x) and g(x) but none seen to work
Any help appreciated !