Internal problem ID [10900]
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 Equation of form
\((a x + b)y''+f(x)y'+g(x)y=0\)
Problem number: 66.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {x y^{\prime \prime }+\left (1-3 n \right ) y^{\prime }-a^{2} n^{2} x^{2 n -1} y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 61
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)
\[ y = c_{1} {\mathrm e}^{a \,x^{n}} \left (-a \,x^{n}+x^{-n} \sqrt {x^{2 n}}\right )+c_{2} {\mathrm e}^{-a \,x^{n}} \left (a \,x^{n}+x^{-n} \sqrt {x^{2 n}}\right ) \]
✓ Solution by Mathematica
Time used: 0.196 (sec). Leaf size: 77
DSolve[x*y''[x]+(1-3*n)*y'[x]-a^2*n^2*x^(2*n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (c_1-\frac {3}{8} i a c_2 \sqrt {x^{2 n}}\right ) \cosh \left (a \sqrt {x^{2 n}}\right )+\frac {1}{8} \left (3 i c_2-8 a c_1 \sqrt {x^{2 n}}\right ) \sinh \left (a \sqrt {x^{2 n}}\right ) \]