- #1
phil ess
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Homework Statement
Find the general solution to x'' + e^(-2t)x = 0, where '' = d2/dt2
Homework Equations
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The Attempt at a Solution
First I did a change of variables: Let u = e^(-t)
Then du/dt = -e^(-t)
dx/dt = dx/du*du/dt = -e^(-t)*dx/du
d2x/dt2 = d/du(dx/dt)du/dt = e^(-2t)*d2x/du2
Subbing into the ODE, I get:
e^(-2t)*d2x/du2 + e^(-2t)x = 0
And I notice that the coefficients are just u^2
(u^2)x'' + (u^2)x = 0
Now at this point I could just cancel out the u^2 and get my sin and cos solutions, but the answer wants Bessel functions, so I use the general solution to ODEs of the form:
x2y'' + x(a+2bxr)y' + [c+dx2s-b(1-a-r)xr+b2x2r]y = 0
with a=0,b=0,c=0,d=1,s=1
The solution is then x(u) = u1/2Zp(u)
With p = 1/2
Or
x(t) = e^(-t/2)[c1*J1/2(e-t+c2*J-1/2(e-t)]
But this isn't the correct answer. There shouldn't be any function of t in front of the bessel functions, and I should be getting Y's and J's of integer order, not just J's of half-integer order.
If anyone has any insights please help!
Thanks