Internal problem ID [11054]
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: 220.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Halm]
\[ \boxed {\left (x^{2}+1\right )^{2} y^{\prime \prime }+a y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 65
dsolve((x^2+1)^2*diff(y(x),x$2)+a*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sqrt {x^{2}+1}\, \left (\frac {x +i}{-x +i}\right )^{\frac {\sqrt {a +1}}{2}}+c_{2} \sqrt {x^{2}+1}\, \left (\frac {x +i}{-x +i}\right )^{-\frac {\sqrt {a +1}}{2}} \]
✓ Solution by Mathematica
Time used: 0.215 (sec). Leaf size: 83
DSolve[(x^2+1)^2*y''[x]+a*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} \sqrt {x^2+1} e^{i \sqrt {a+1} \arctan (x)} \left (\frac {i c_2 (1-i x)^{\sqrt {a+1}} (1+i x)^{-\sqrt {a+1}}}{\sqrt {a+1}}+2 c_1\right ) \]