61.32.12 problem 222 A

Internal problem ID [12643]
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 : 222 A
Date solved : Monday, March 31, 2025 at 06:49:07 AM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 75

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

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

✓ Mathematica. Time used: 0.181 (sec). Leaf size: 104

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

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

✓ Sympy. Time used: 0.443 (sec). Leaf size: 124

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

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