- #1
the0
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Homework Statement
Find the set of functions from [itex](-1,1)→ℝ[/itex] which are solutions of:
[itex](x^{2}-1)\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}-4y = 0[/itex]
Homework Equations
The Attempt at a Solution
There is a hint which says to use the change of variable:
[itex]x=cos(θ)[/itex]
doing this I get:
[itex]\frac{dx}{dθ} = -sin(θ)[/itex]
[itex]\Rightarrow[/itex]
[itex]dx = -sin(θ)dθ[/itex]
[itex]\Rightarrow[/itex]
[itex](a): \frac{dy}{dx} = \frac{dy}{dθ}\frac{1}{-sin(θ)}[/itex]
[itex](b): \frac{d^{2}y}{dx^{2}} = \frac{d^{2}y}{dθ^{2}}\frac{1}{sin^{2}(θ)}[/itex]
Can I do this?!?
If so, substituting everything in gives:
[itex]-sin^{2}(θ)\frac{d^{2}y}{dθ^{2}}\frac{1}{sin^{2}(θ)} + cos(θ)\frac{dy}{dθ}\frac{1}{-sin(θ)} - 4y = 0[/itex]
[itex]\Rightarrow[/itex]
[itex]y'' + cot(θ)y' + 4y = 0[/itex]
Now... I am not sure.
Have I made some mistake?
Or should I be able to solve this?
Could someone please point me in the right direction?
Thanks a lot!