61.32.14 problem 223

Internal problem ID [12724]
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 : Tuesday, January 28, 2025 at 04:16:44 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*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 55

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

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

✓ Solution by Mathematica

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

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

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