Internal problem ID [11024]
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 Equation of form
\((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 190.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) y^{\prime }+s x y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 203
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)
\[ y \left (x \right ) = c_{1} x^{\frac {a -c}{a}} \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-2 b -4 s +1}}{4}, -\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-2 b -4 s +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )+c_{2} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-2 b -4 s +1}\, a +a b -3 a -2 c}{4 a}, \frac {\sqrt {b^{2}-2 b -4 s +1}\, a +\left (-b +3\right ) a +2 c}{4 a}\right ], \left [\frac {a +c}{2 a}\right ], -\frac {x^{2}}{a}\right ) \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \]
✓ Solution by Mathematica
Time used: 0.967 (sec). Leaf size: 185
DSolve[x*(x^2+a)*y''[x]+(b*x^2+c)*y'[x]+s*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 a^{\frac {1}{2} \left (\frac {c}{a}-1\right )} x^{1-\frac {c}{a}} \operatorname {Hypergeometric2F1}\left (\frac {a \left (b+\sqrt {b^2-2 b-4 s+1}+1\right )-2 c}{4 a},\frac {b a-\sqrt {b^2-2 b-4 s+1} a+a-2 c}{4 a},\frac {3}{2}-\frac {c}{2 a},-\frac {x^2}{a}\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2-2 b-4 s+1}-1\right ),\frac {1}{4} \left (b+\sqrt {b^2-2 b-4 s+1}-1\right ),\frac {a+c}{2 a},-\frac {x^2}{a}\right ) \]