Internal problem ID [10880]
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 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 46.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+2 a \,x^{n} y^{\prime }+a \left (a \,x^{2 n}+n \,x^{-1+n}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 38
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)
\[ y = c_{1} {\mathrm e}^{-\frac {a \,x^{n +1}}{n +1}}+c_{2} {\mathrm e}^{-\frac {a \,x^{n +1}}{n +1}} x \]
✓ Solution by Mathematica
Time used: 0.112 (sec). Leaf size: 28
DSolve[y''[x]+2*a*x^n*y'[x]+a*(a*x^(2*n)+n*x^(n-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (c_2 x+c_1) e^{-\frac {a x^{n+1}}{n+1}} \]