61.33.18 problem 256

Internal problem ID [12756]
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 : 256
Date solved : Tuesday, January 28, 2025 at 08:24:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 1.271 (sec). Leaf size: 53

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

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

✓ Solution by Mathematica

Time used: 60.196 (sec). Leaf size: 106

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

\[ y(x)\to \exp \left (-\frac {x \left ((a d-b c) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {a x^n}{b}\right )+b c\right )}{a b}\right ) \left (\int _1^x\exp \left (\frac {\left (b c+(a d-b c) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {a K[1]^n}{b}\right )\right ) K[1]}{a b}\right ) c_1dK[1]+c_2\right ) \]