61.33.14 problem 252

Internal problem ID [12752]
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
Date solved : Tuesday, January 28, 2025 at 04:17:48 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} 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 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 59

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)
 

\[ y = c_{1} {\mathrm e}^{i b \left (\int x^{n -1} \sqrt {-\frac {1}{x^{2 n}+a}}d x \right )}+c_{2} {\mathrm e}^{-i b \left (\int x^{n -1} \sqrt {-\frac {1}{x^{2 n}+a}}d x \right )} \]

✓ Solution by Mathematica

Time used: 1.274 (sec). Leaf size: 47

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

\[ y(x)\to c_1 \cosh \left (\frac {b \text {arcsinh}\left (\frac {x^n}{\sqrt {a}}\right )}{n}\right )+i c_2 \sinh \left (\frac {b \text {arcsinh}\left (\frac {x^n}{\sqrt {a}}\right )}{n}\right ) \]