60.3.327 problem 1333

Internal problem ID [11337]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1333
Date solved : Tuesday, January 28, 2025 at 06:02:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}+\frac {v \left (v +1\right ) y}{4 x^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.348 (sec). Leaf size: 74

dsolve(diff(diff(y(x),x),x) = -1/2/x*(3*x-1)/(x-1)*diff(y(x),x)+1/4*v*(v+1)/x^2*y(x),y(x), singsol=all)
 

\[ y = \frac {\left (c_{2} \Gamma \left (v +\frac {1}{2}\right )^{2} \left (v +\frac {1}{2}\right ) \operatorname {LegendreP}\left (-\frac {1}{2}, -v -\frac {1}{2}, \frac {-x -1}{x -1}\right )+\sec \left (\pi v \right ) \operatorname {LegendreP}\left (-\frac {1}{2}, v +\frac {1}{2}, \frac {-x -1}{x -1}\right ) \pi c_{1} \right ) x^{{1}/{4}}}{\sqrt {1-x}\, \Gamma \left (v +\frac {1}{2}\right )} \]

✓ Solution by Mathematica

Time used: 0.141 (sec). Leaf size: 70

DSolve[D[y[x],{x,2}] == (v*(1 + v)*y[x])/(4*x^2) - ((-1 + 3*x)*D[y[x],x])/(2*(-1 + x)*x),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 i^{-v} x^{-v/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-v,\frac {1}{2}-v,x\right )+c_2 i^{v+1} x^{\frac {v+1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},v+1,v+\frac {3}{2},x\right ) \]