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Peregrine

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**Simple Diff Eq Help**

I am trying to solve the following Diff Eq:

[tex]\frac{d^2x}{dy^2}+(\frac{y}{2}-\frac{1}{y})\frac{dx}{dy}=0[/tex]

I tried to solve by setting [tex]\frac{dx}{dy}=z[/tex]

so: [tex]\frac{dz}{dy}+(\frac{y}{2}-\frac{1}{y})z=0[/tex]

I know the general solution to this is:

[tex]z=-e^{-\int{\frac{y}{2}-\frac{1}{y}dy}}\int{0}dy[/tex]

This then yields:

[tex]z=-C_1e^{ln(y)-1/4y^2}=\frac{dx}{dy}[/tex]

And trying to integrate again, Using u-substitution, [tex]u = ln(y) -1/4y^2[/tex]

[tex]du=(\frac{1}{y} - \frac{y}{2}) dy[/tex]

[tex]dy = \frac{2}{2-y^2} du[/tex]

Now, can I leave that y tern in the u-substitution? Or did I make a mistake along the way?

Also, is there an easier way to solve this integral than the path I've taken? Thanks.

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