61.31.9 problem 190

Internal problem ID [12690]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 190
Date solved : Tuesday, January 28, 2025 at 08:11:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.477 (sec). Leaf size: 175

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

\[ y = \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \left (x^{\frac {-c +a}{a}} \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-2 b -4 s +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-2 b -4 s +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right ) c_{1} +\operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {3}{4}+\frac {c}{2 a}+\frac {\sqrt {b^{2}-2 b -4 s +1}}{4}, -\frac {\sqrt {b^{2}-2 b -4 s +1}}{4}-\frac {b}{4}+\frac {3}{4}+\frac {c}{2 a}\right ], \left [\frac {1}{2}+\frac {c}{2 a}\right ], -\frac {x^{2}}{a}\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.582 (sec). Leaf size: 185

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

\[ y(x)\to c_2 a^{\frac {1}{2} \left (\frac {c}{a}-1\right )} x^{1-\frac {c}{a}} \operatorname {Hypergeometric2F1}\left (\frac {a \left (b+\sqrt {b^2-2 b-4 s+1}+1\right )-2 c}{4 a},\frac {b a-\sqrt {b^2-2 b-4 s+1} a+a-2 c}{4 a},\frac {3}{2}-\frac {c}{2 a},-\frac {x^2}{a}\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2-2 b-4 s+1}-1\right ),\frac {1}{4} \left (b+\sqrt {b^2-2 b-4 s+1}-1\right ),\frac {a+c}{2 a},-\frac {x^2}{a}\right ) \]