55.33.14 problem 223

Internal problem ID [13997]
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
Problem number : 223
Date solved : Thursday, October 02, 2025 at 09:08:34 AM
CAS classification : [_Halm]

\begin{align*} 4 \left (x^{2}+1\right )^{2} y^{\prime \prime }+\left (a \,x^{2}+a -3\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 55

ode:=4*(x^2+1)^2*diff(diff(y(x),x),x)+(a*x^2+a-3)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (x^{2}+1\right )^{{1}/{4}} \left (\left (x +\sqrt {x^{2}+1}\right )^{-\frac {\sqrt {-a +1}}{2}} c_2 +\left (x +\sqrt {x^{2}+1}\right )^{\frac {\sqrt {-a +1}}{2}} c_1 \right ) \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 70

ode=4*(x^2+1)^2*D[y[x],{x,2}]+(a*x^2+a-3)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} 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 ) \end{align*}

✓ Sympy. Time used: 0.269 (sec). Leaf size: 116

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(4*(x**2 + 1)**2*Derivative(y(x), (x, 2)) + (a*x**2 + a - 3)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\sqrt [4]{x^{2} + 1} \left (C_{1} \sqrt {\frac {x^{2}}{x^{2} + 1}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2} - \frac {\sqrt {1 - a}}{4}, 1 - \frac {\sqrt {1 - a}}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {x^{2}}{x^{2} + 1}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {\sqrt {1 - a}}{4}, \frac {1}{2} - \frac {\sqrt {1 - a}}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {x^{2}}{x^{2} + 1}} \right )}\right ) \sqrt [4]{x^{2}} e^{\frac {\sqrt {1 - a} \log {\left (x^{2} + 1 \right )}}{4}}}{\sqrt {x}} \]