60.3.383 problem 1389

Internal problem ID [11393]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1389
Date solved : Tuesday, January 28, 2025 at 06:04:50 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 )^{2}-4 n^{2} x \right ) y}{4 x^{2} \left (x -1\right )^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.234 (sec). Leaf size: 87

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)^2-4*n^2*x)/x^2/(x-1)^2*y(x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.985 (sec). Leaf size: 109

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

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