Internal problem ID [11045]
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: 221.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (x^{2}-1\right )^{2} y^{\prime \prime }+a y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 55
dsolve((x^2-1)^2*diff(y(x),x$2)+a*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \sqrt {x^{2}-1}\, \left (\left (\frac {-1+x}{1+x}\right )^{\frac {\sqrt {-a +1}}{2}} c_{1} +\left (\frac {-1+x}{1+x}\right )^{-\frac {\sqrt {-a +1}}{2}} c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.249 (sec). Leaf size: 88
DSolve[(x^2-1)^2*y''[x]+a*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\left (1-x^2\right )^{\frac {1}{2}-\frac {\sqrt {1-a}}{2}} \left (2 \sqrt {1-a} c_1 (1-x)^{\sqrt {1-a}}+c_2 (x+1)^{\sqrt {1-a}}\right )}{2 \sqrt {1-a}} \]