61.33.26 problem 264

Internal problem ID [12764]
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 : 264
Date solved : Tuesday, January 28, 2025 at 04:18:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 41

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

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

✓ Solution by Mathematica

Time used: 60.096 (sec). Leaf size: 116

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

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