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- Summary:
- How does one go from ##\phi## to ##x=\cos(\phi)##

I'm reading "Differential Equations with Applications and Historical Notes" by George F. Simmons and I am confused about something on pages 431-432

He has the second order ordinary differential equation

$$\frac {d^2v} {d\phi^2} + \frac {\cos(\phi)} {\sin(\phi)} \frac {dv} {d\phi} + n(n+1)v = 0 ~~~~~~~~~~~~~~~~~~~ eq. 1$$

And then using a change of independent variable from ## \phi ## to ## x = \cos(\phi) ## eq 1 is transformed into the Legendre equation

$$ (1-x^2) \frac {d^2v} {dx^2} - 2x \frac {dv} {dx} + n(n+1)v = 0 ~~~~~~~~~~~~ eq. 2 $$

But I don't see how he got from eq 1 to eq 2

Anyone feel like helping me out?

He has the second order ordinary differential equation

$$\frac {d^2v} {d\phi^2} + \frac {\cos(\phi)} {\sin(\phi)} \frac {dv} {d\phi} + n(n+1)v = 0 ~~~~~~~~~~~~~~~~~~~ eq. 1$$

And then using a change of independent variable from ## \phi ## to ## x = \cos(\phi) ## eq 1 is transformed into the Legendre equation

$$ (1-x^2) \frac {d^2v} {dx^2} - 2x \frac {dv} {dx} + n(n+1)v = 0 ~~~~~~~~~~~~ eq. 2 $$

But I don't see how he got from eq 1 to eq 2

Anyone feel like helping me out?