Internal problem ID [10927]
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: 93.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+\left (x^{n}+1-n \right ) y^{\prime }+b \,x^{2 n -1} y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 61
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)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x^{n}}{2 n}} \sinh \left (\frac {x^{n} \sqrt {\frac {-4 b +1}{n^{2}}}}{2}\right )+c_{2} {\mathrm e}^{-\frac {x^{n}}{2 n}} \cosh \left (\frac {x^{n} \sqrt {\frac {-4 b +1}{n^{2}}}}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.094 (sec). Leaf size: 53
DSolve[x*y''[x]+(x^n+1-n)*y'[x]+b*x^(2*n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {\left (\sqrt {1-4 b}+1\right ) x^n}{2 n}} \left (c_2 e^{\frac {\sqrt {1-4 b} x^n}{n}}+c_1\right ) \]