61.29.12 problem 121

Internal problem ID [12621]
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 : 121
Date solved : Tuesday, January 28, 2025 at 08:02:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (a \,x^{3 n}+b \,x^{2 n}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 1.168 (sec). Leaf size: 177

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

\[ y = \frac {x^{-\frac {n}{2}+\frac {1}{2}} \left (\frac {2 \,3^{{5}/{6}} \pi c_{2} \left (a \,x^{n}+b \right ) \operatorname {BesselI}\left (\frac {1}{3}, \frac {2 \sqrt {\frac {-a^{3} x^{3 n}-3 x^{2 n} a^{2} b -3 x^{n} a \,b^{2}-b^{3}}{n^{2} a^{2}}}}{3}\right )}{3}+c_{1} \operatorname {BesselI}\left (-\frac {1}{3}, \frac {2 \sqrt {\frac {-a^{3} x^{3 n}-3 x^{2 n} a^{2} b -3 x^{n} a \,b^{2}-b^{3}}{n^{2} a^{2}}}}{3}\right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{{2}/{3}} {\left (-\frac {\left (a \,x^{n}+b \right )^{3}}{n^{2} a^{2}}\right )}^{{1}/{3}}\right )}{3 {\left (-\frac {\left (a \,x^{n}+b \right )^{3}}{n^{2} a^{2}}\right )}^{{1}/{6}} \Gamma \left (\frac {2}{3}\right )} \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

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

Not solved