Internal problem ID [11082]
Internal file name [OUTPUT/10339_Wednesday_January_24_2024_10_18_01_PM_37857849/index.tex]
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.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {x^{2} \left (x^{n} a +b \right )^{2} y^{\prime \prime }+\left (1+n \right ) x \left (x^{2 n} a^{2}-b^{2}\right ) y^{\prime }+c y=0} \]
Maple trace Kovacic algorithm successful
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Group is reducible or imprimitive <- Kovacics algorithm successful <- Equivalence, under non-integer power transformations successful`
✓ Solution by Maple
Time used: 0.172 (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)
\[ y \left (x \right ) = \sqrt {a \,x^{2 n}+b \,x^{n}}\, \left (a \,x^{n}+b \right )^{\frac {-n -1}{n}} x \left (\left (\frac {x^{n}}{a \,x^{n}+b}\right )^{-\frac {\sqrt {\frac {\left (n +2\right )^{2} b^{2}-4 c}{n^{2} a^{2}}}\, a}{2 b}} c_{2} +\left (\frac {x^{n}}{a \,x^{n}+b}\right )^{\frac {\sqrt {\frac {\left (n +2\right )^{2} b^{2}-4 c}{n^{2} a^{2}}}\, a}{2 b}} c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.407 (sec). Leaf size: 149
DSolve[x^2*(a*x^n+b)^2*y''[x]+(n+1)*x*(a^2*x^(2*n)-b^2)*y'[x]+c*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \exp \left (\frac {\left (b (n+2)-\sqrt {c} \sqrt {\frac {b^2 (n+2)^2-4 c}{c}}\right ) \left (-\log \left (a x^n+b\right )-\log (b)+n \log (x)-\log (n)\right )}{2 b n}\right )+c_2 \exp \left (\frac {\left (\sqrt {c} \sqrt {\frac {b^2 (n+2)^2-4 c}{c}}+b (n+2)\right ) \left (-\log \left (a x^n+b\right )-\log (b)+n \log (x)-\log (n)\right )}{2 b n}\right ) \]