How to Represent an Almost Kummer's Equation in Terms of Kummer's or Solve It?

[intervoxel]

I met the following equation in my research, which is almost Kummer's equation (without the 2):

x*y''+(b-2*x)*y'-a*y=0

How can I represent this equation in terms of Kummer's? Or else, how solve it?

 

[Matthew Rodman]

Take your equation, and make the change of variable

$$\tau = 2 x$$

This means that

$$y^{\prime}_{x} = 2 y^{\prime}_{\tau}$$

and

$$y^{\prime \prime}_{xx} = 4 y^{\prime \prime}_{\tau \tau}$$

Substitute these into your equation, and it becomes

$$\tau y^{\prime \prime}_{\tau \tau} + (b - \tau) y^{\prime}_{\tau} - \frac{a}{2} y = 0$$

which is the hypergeomtric equation in the new variable.

 

[intervoxel]

Perfect! Thank you.