61.32.8 problem 218

Internal problem ID [12718]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-7
Problem number : 218
Date solved : Tuesday, January 28, 2025 at 08:23:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} a \,x^{2} \left (x -1\right )^{2} y^{\prime \prime }+\left (b \,x^{2}+c x +d \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.375 (sec). Leaf size: 267

dsolve(a*x^2*(x-1)^2*diff(y(x),x$2)+(b*x^2+c*x+d)*y(x)=0,y(x), singsol=all)
 

\[ y = \left (x -1\right )^{\frac {\sqrt {a}-\sqrt {a -4 b -4 c -4 d}}{2 \sqrt {a}}} \left (c_{2} x^{-\frac {-\sqrt {a}+\sqrt {a -4 d}}{2 \sqrt {a}}} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {a -4 b -4 c -4 d}-\sqrt {a}+\sqrt {a -4 d}-\sqrt {a -4 b}}{2 \sqrt {a}}, -\frac {\sqrt {a -4 b -4 c -4 d}-\sqrt {a}+\sqrt {a -4 d}+\sqrt {a -4 b}}{2 \sqrt {a}}\right ], \left [1-\frac {\sqrt {a -4 d}}{\sqrt {a}}\right ], x\right )+c_{1} x^{\frac {\sqrt {a -4 d}+\sqrt {a}}{2 \sqrt {a}}} \operatorname {hypergeom}\left (\left [\frac {-\sqrt {a -4 b -4 c -4 d}+\sqrt {a}+\sqrt {a -4 d}+\sqrt {a -4 b}}{2 \sqrt {a}}, \frac {-\sqrt {a -4 b -4 c -4 d}+\sqrt {a}+\sqrt {a -4 d}-\sqrt {a -4 b}}{2 \sqrt {a}}\right ], \left [1+\frac {\sqrt {a -4 d}}{\sqrt {a}}\right ], x\right )\right ) \]

✓ Solution by Mathematica

Time used: 125.756 (sec). Leaf size: 413606

DSolve[a*x^2*(x-1)^2*D[y[x],{x,2}]+(b*x^2+c*x+d)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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