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The following is an explanation from my tutorial. I do not understand it.

[tex]{\frac{d}{dx}(y_1{{y_2}'}-y_2{{y_1}'})+P(x)(y_1{{y_2}'}-y_2{{y_1}'})=0[/tex]

Overlooking for the moment that P(x) may be undefined at certain values of x(so-called-singular points of the equation), we recognize this equation to be a separable first-order differential equation for the function [tex]y_1{{y_2}'}-y_2{{y_1}'}[/tex] that can be integrated to give:

[tex]y_1{{y_2}'}-y_2{{y_1}'}= C_{12}(exp(-{\int{P(x')dx'}})[/tex]

[tex]y_1{{y_2}'}-y_2{{y_1}'}= C_{12}\phi[P][/tex]

Where [tex]C_{12}[/tex] is an integration constant that depends, possibly, on the choices of functions y_1 and y_2, and phi[P] is a functional, that is, a function that depends upon another function, in this case, P(x). The expression on the left hand side is called the Wronskian of the functions y_1 and y_2.

I do not understand how they separate this equation. Here is what I do. I first separate like this:

[tex]{\frac{d}{dx}(y_1{{y_2}'}-y_2{{y_1}'})=-P(x)(y_1{{y_2}'}-y_2{{y_1}'})[/tex]

Then I multiply both sides by dx. Next I divide both sides by [tex]y_1{{y_2}'}-y_2{{y_1}'}[/tex]

Then I have:

[tex]{\int{1}d}=-{\int{P(x)dx}[/tex]

What am I doing wrong here?

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# Homework Help: Separable FODE

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