Internal problem ID [11042]
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: 218.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {a \,x^{2} \left (x -1\right )^{2} y^{\prime \prime }+\left (b \,x^{2}+c x +d \right ) y=0} \]
✓ Solution by Maple
Time used: 0.125 (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)
\[ y \left (x \right ) = \left (-1+x \right )^{-\frac {\sqrt {a -4 b -4 c -4 d}-\sqrt {a}}{2 \sqrt {a}}} \left (c_{1} x^{\frac {\sqrt {a}+\sqrt {a -4 d}}{2 \sqrt {a}}} \operatorname {hypergeom}\left (\left [\frac {-\sqrt {a -4 b -4 c -4 d}+\sqrt {a}+\sqrt {a -4 d}+\sqrt {a -4 b}}{2 \sqrt {a}}, -\frac {\sqrt {a -4 b -4 c -4 d}-\sqrt {a}-\sqrt {a -4 d}+\sqrt {a -4 b}}{2 \sqrt {a}}\right ], \left [1+\frac {\sqrt {a -4 d}}{\sqrt {a}}\right ], x\right )+c_{2} x^{-\frac {-\sqrt {a}+\sqrt {a -4 d}}{2 \sqrt {a}}} \operatorname {hypergeom}\left (\left [\frac {-\sqrt {a -4 b -4 c -4 d}+\sqrt {a}-\sqrt {a -4 d}+\sqrt {a -4 b}}{2 \sqrt {a}}, -\frac {\sqrt {a -4 b -4 c -4 d}-\sqrt {a}+\sqrt {a -4 d}+\sqrt {a -4 b}}{2 \sqrt {a}}\right ], \left [1-\frac {\sqrt {a -4 d}}{\sqrt {a}}\right ], x\right )\right ) \]
✓ Solution by Mathematica
Time used: 135.53 (sec). Leaf size: 413606
DSolve[a*x^2*(x-1)^2*y''[x]+(b*x^2+c*x+d)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
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