61.29.18 problem 127

Internal problem ID [12627]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 127
Date solved : Tuesday, January 28, 2025 at 03:23:30 AM
CAS classification : [[_Bessel, _modified]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 15

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

\[ y = c_{1} \operatorname {BesselI}\left (\nu , x\right )+c_{2} \operatorname {BesselK}\left (\nu , x\right ) \]

✓ Solution by Mathematica

Time used: 0.034 (sec). Leaf size: 34

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

\[ y(x)\to c_1 \operatorname {BesselJ}\left (\sqrt {\nu },-i x\right )+c_2 \operatorname {BesselY}\left (\sqrt {\nu },-i x\right ) \]