61.28.6 problem 66

Internal problem ID [12566]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 66
Date solved : Tuesday, January 28, 2025 at 08:02:21 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Solution by Maple

Time used: 0.316 (sec). Leaf size: 62

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

\[ y = c_{2} {\mathrm e}^{-a \,x^{n}} \left (a \,x^{n}+x^{-n} \sqrt {x^{2 n}}\right )-{\mathrm e}^{a \,x^{n}} c_{1} \left (a \,x^{n}-x^{-n} \sqrt {x^{2 n}}\right ) \]

✓ Solution by Mathematica

Time used: 0.082 (sec). Leaf size: 77

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

\[ y(x)\to \left (c_1-\frac {3}{8} i a c_2 \sqrt {x^{2 n}}\right ) \cosh \left (a \sqrt {x^{2 n}}\right )+\frac {1}{8} \left (3 i c_2-8 a c_1 \sqrt {x^{2 n}}\right ) \sinh \left (a \sqrt {x^{2 n}}\right ) \]