61.33.21 problem 259

Internal problem ID [12759]
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 : 259
Date solved : Tuesday, January 28, 2025 at 08:24:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (a \,x^{n}+b \right )^{2} y^{\prime \prime }+\left (n +1\right ) x \left (a^{2} x^{2 n}-b^{2}\right ) y^{\prime }+c y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.514 (sec). Leaf size: 127

dsolve(x^2*(a*x^n+b)^2*diff(y(x),x$2)+(n+1)*x*(a^2*x^(2*n)-b^2)*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
 

\[ y = \sqrt {a \,x^{2 n}+b \,x^{n}}\, \left (a \,x^{n}+b \right )^{\frac {-n -1}{n}} x \left (\left (\frac {x^{n}}{a \,x^{n}+b}\right )^{-\frac {\sqrt {\frac {\left (n +2\right )^{2} b^{2}-4 c}{a^{2} n^{2}}}\, a}{2 b}} c_{2} +\left (\frac {x^{n}}{a \,x^{n}+b}\right )^{\frac {\sqrt {\frac {\left (n +2\right )^{2} b^{2}-4 c}{a^{2} n^{2}}}\, a}{2 b}} c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.687 (sec). Leaf size: 149

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

\[ y(x)\to c_1 \exp \left (\frac {\left (b (n+2)-\sqrt {c} \sqrt {\frac {b^2 (n+2)^2-4 c}{c}}\right ) \left (-\log \left (a x^n+b\right )-\log (b)+n \log (x)-\log (n)\right )}{2 b n}\right )+c_2 \exp \left (\frac {\left (\sqrt {c} \sqrt {\frac {b^2 (n+2)^2-4 c}{c}}+b (n+2)\right ) \left (-\log \left (a x^n+b\right )-\log (b)+n \log (x)-\log (n)\right )}{2 b n}\right ) \]