61.32.12 problem 222 A

Internal problem ID [12722]
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 : Tuesday, January 28, 2025 at 04:16:40 AM
CAS classification : [[_Emden, _Fowler]]

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 75

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

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

✓ Solution by Mathematica

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

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

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