Internal problem ID [11058]
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: 223.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Halm]
\[ \boxed {4 \left (x^{2}+1\right )^{2} y^{\prime \prime }+\left (a \,x^{2}+a -3\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 61
dsolve(4*(x^2+1)^2*diff(y(x),x$2)+(a*x^2+a-3)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (x^{2}+1\right )^{\frac {1}{4}} \left (x +\sqrt {x^{2}+1}\right )^{\frac {\sqrt {-a +1}}{2}}+c_{2} \left (x^{2}+1\right )^{\frac {1}{4}} \left (x +\sqrt {x^{2}+1}\right )^{-\frac {\sqrt {-a +1}}{2}} \]
✓ Solution by Mathematica
Time used: 0.067 (sec). Leaf size: 70
DSolve[4*(x^2+1)^2*y''[x]+(a*x^2+a-3)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sqrt {x^2+1} \left (c_1 P_{\frac {1}{2} \left (\sqrt {1-a}-1\right )}^{\frac {1}{2}}(i x)+c_2 Q_{\frac {1}{2} \left (\sqrt {1-a}-1\right )}^{\frac {1}{2}}(i x)\right ) \]