61.27.36 problem 46

Internal problem ID [12546]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 46
Date solved : Tuesday, January 28, 2025 at 03:21:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 24

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

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

✓ Solution by Mathematica

Time used: 0.067 (sec). Leaf size: 28

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

\[ y(x)\to (c_2 x+c_1) e^{-\frac {a x^{n+1}}{n+1}} \]