61.32.10 problem 220

Internal problem ID [12641]
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 : 220
Date solved : Monday, March 31, 2025 at 06:49:03 AM
CAS classification : [_Halm]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 55

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

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

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 86

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

\[ y(x)\to \exp \left (\int _1^x\frac {K[1]+i \sqrt {a+1}}{K[1]^2+1}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {K[1]+i \sqrt {a+1}}{K[1]^2+1}dK[1]\right )dK[2]+c_1\right ) \]

✓ Sympy. Time used: 0.353 (sec). Leaf size: 97

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

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