14.In several problems in mathematical physics, it is necessary to study the differential equation (25) x(1−x)y′′+(γ−(1+α+β)x)y′−αβy=0,x1−xy″+γ−1+α+βxy′−αβy=0, where α, β, and γ are constants. This equation is known as the hypergeometric equation. a.Show that x = 0 is a regular singular point and that the roots of the indicial equation are 0 and 1 − γ. b.Show that x = 1 is a regular singular point and that the roots of the indicial equation are 0 and γ − α − β. c.Assuming that 1 − γ is not a positive integer, show that, in the neighborhood of x = 0, one solution of equation (25) is y1(x)=1+αβγ⋅1!x+α(α+1)β(β+1)γ(γ+1)2!x2+⋯.y1x=1+αβγ·1!x+αα+1ββ+1γγ+12!x2+⋯. What would you expect the radius of convergence of this series to be? d.Assuming that 1 − γ is not an integer or zero, show that a second solution for 0 < x < 1 is y2(x)=x1−γ(1+(α−γ+1)(β−γ+1)(2−γ)1!x+(α−γ+1)(α−γ+2)(β−γ+1)(β−γ+2)(2−γ)(3−γ)2!x2+⋯).y2x=x1−γ1+α−γ+1β−γ+12−γ1!x+α−γ+1α−γ+2β−γ+1β−γ+22−γ3−γ2!x2+⋯. e.Show that the point at infinity is a regular singular point and that the roots of the indicial equation are α and β. See Problem 32 of Section 5.4.
14.In several problems in mathematical physics, it is necessary to study the differential equation
where α, β, and γ are constants. This equation is known as the hypergeometric equation.
a.Show that x = 0 is a regular singular point and that the roots of the indicial equation are 0 and 1 − γ.
b.Show that x = 1 is a regular singular point and that the roots of the indicial equation are 0 and γ − α − β.
c.Assuming that 1 − γ is not a positive integer, show that, in the neighborhood of x = 0, one solution of equation (25) is
What would you expect the radius of convergence of this series to be?
d.Assuming that 1 − γ is not an integer or zero, show that a second solution for 0 < x < 1 is
e.Show that the point at infinity is a regular singular point and that the roots of the indicial equation are α and β. See Problem 32 of Section 5.4.
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 4 images