Internal problem ID [11052]
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 Equation of form
\((a_4 x^4+a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 227.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {a \left (x^{2}-1\right )^{2} y^{\prime \prime }+b x \left (x^{2}-1\right ) y^{\prime }+\left (c \,x^{2}+d x +e \right ) y=0} \]
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 562
dsolve(a*(x^2-1)^2*diff(y(x),x$2)+b*x*(x^2-1)*diff(y(x),x)+(c*x^2+d*x+e)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (x^{2}-1\right )^{-\frac {b}{4 a}} \sqrt {2+2 x}\, \left (c_{1} \operatorname {hypergeom}\left (\left [-\frac {-\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}+2 \sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}+\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}-2 a}{4 a}, \frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}+2 \sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}-\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}\right ], \left [-\frac {-2 a +\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {1}{2}+\frac {x}{2}\right ) \left (\frac {1}{2}+\frac {x}{2}\right )^{-\frac {\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}}+c_{2} \operatorname {hypergeom}\left (\left [\frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}+2 \sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}+\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}, \frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}-2 \sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}+\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}+2 a}{4 a}\right ], \left [\frac {2 a +\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{2 a}\right ], \frac {1}{2}+\frac {x}{2}\right ) \left (\frac {1}{2}+\frac {x}{2}\right )^{\frac {\sqrt {4 a^{2}+\left (-4 b -4 c +4 d -4 e \right ) a +b^{2}}}{4 a}}\right ) \sqrt {2 x -2}\, \left (-\frac {1}{2}+\frac {x}{2}\right )^{\frac {\sqrt {4 a^{2}+\left (-4 b -4 c -4 d -4 e \right ) a +b^{2}}}{4 a}}}{4} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[a*(x^2-1)^2*y''[x]+b*x*(x^2-1)*y'[x]+(c*x^2+d*x+e)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Timed out