60.3.382 problem 1388

Internal problem ID [11392]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1388
Date solved : Tuesday, January 28, 2025 at 06:04:49 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 {\left (v \left (v +1\right ) \left (x -1\right )-a^{2} x \right ) y}{4 x^{2} \left (x -1\right )^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.236 (sec). Leaf size: 77

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-1)-a^2*x)/x^2/(x-1)^2*y(x),y(x), singsol=all)
 

\[ y = \left (x -1\right )^{-\frac {a}{2}} \left (\operatorname {hypergeom}\left (\left [-\frac {v}{2}-\frac {a}{2}, \frac {1}{2}-\frac {v}{2}-\frac {a}{2}\right ], \left [\frac {1}{2}-v \right ], x\right ) x^{-\frac {v}{2}} c_{1} +c_{2} \sqrt {x}\, x^{\frac {v}{2}} \operatorname {hypergeom}\left (\left [1+\frac {v}{2}-\frac {a}{2}, \frac {1}{2}+\frac {v}{2}-\frac {a}{2}\right ], \left [\frac {3}{2}+v \right ], x\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.840 (sec). Leaf size: 127

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

\[ y(x)\to (-1)^{-v} (x-1)^{\frac {a+1}{2}} x^{\frac {1}{4}-\frac {v}{2}} e^{-\frac {1}{4} \int \frac {1-3 x}{x-x^2} \, dx} \left (c_1 (-1)^v x^{v+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (a+v+1),\frac {1}{2} (a+v+2),v+\frac {3}{2},x\right )-i c_2 \operatorname {Hypergeometric2F1}\left (\frac {a-v}{2},\frac {1}{2} (a-v+1),\frac {1}{2}-v,x\right )\right ) \]