61.28.5 problem 65

Internal problem ID [12565]
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 : 65
Date solved : Tuesday, January 28, 2025 at 03:21:46 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x y^{\prime \prime }+n y^{\prime }+b \,x^{1-2 n} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 43

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

\[ y = c_{1} \sin \left (\frac {x^{-n +1} \sqrt {b}}{n -1}\right )+c_{2} \cos \left (\frac {x^{-n +1} \sqrt {b}}{n -1}\right ) \]

✓ Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 52

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

\[ y(x)\to c_1 \cos \left (\frac {\sqrt {b} x^{1-n}}{n-1}\right )+c_2 \sin \left (\frac {\sqrt {b} x^{1-n}}{1-n}\right ) \]