61.28.33 problem 93

Internal problem ID [12593]
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 : 93
Date solved : Tuesday, January 28, 2025 at 08:02:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.185 (sec). Leaf size: 53

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

\[ y = {\mathrm e}^{-\frac {x^{n}}{2 n}} \left (c_{1} \sinh \left (\frac {x^{n} \sqrt {\frac {1-4 b}{n^{2}}}}{2}\right )+c_{2} \cosh \left (\frac {x^{n} \sqrt {\frac {1-4 b}{n^{2}}}}{2}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.056 (sec). Leaf size: 53

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

\[ y(x)\to e^{-\frac {\left (\sqrt {1-4 b}+1\right ) x^n}{2 n}} \left (c_2 e^{\frac {\sqrt {1-4 b} x^n}{n}}+c_1\right ) \]