61.32.10 problem 220

Internal problem ID [12720]
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 : Tuesday, January 28, 2025 at 04:16:36 AM
CAS classification : [_Halm]

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 55

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

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

✓ Solution by Mathematica

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

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

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