Internal problem ID [11086]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-8. Other equations.
Problem number: 252.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {x \left (x^{2 n}+a \right ) y^{\prime \prime }+\left (x^{2 n}+a -a n \right ) y^{\prime }-b^{2} x^{2 n -1} y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 61
dsolve(x*(x^(2*n)+a)*diff(y(x),x$2)+(x^(2*n)+a-a*n)*diff(y(x),x)-b^2*x^(2*n-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{\int i b \,x^{n -1} \sqrt {-\frac {1}{x^{2 n}+a}}d x}+c_{2} {\mathrm e}^{-\left (\int i b \,x^{n -1} \sqrt {-\frac {1}{x^{2 n}+a}}d x \right )} \]
✓ Solution by Mathematica
Time used: 0.458 (sec). Leaf size: 47
DSolve[x*(x^(2*n)+a)*y''[x]+(x^(2*n)+a-a*n)*y'[x]-b^2*x^(2*n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \cosh \left (\frac {b \text {arcsinh}\left (\frac {x^n}{\sqrt {a}}\right )}{n}\right )+i c_2 \sinh \left (\frac {b \text {arcsinh}\left (\frac {x^n}{\sqrt {a}}\right )}{n}\right ) \]