61.28.34 problem 94

Internal problem ID [12594]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 94
Date solved : Tuesday, January 28, 2025 at 03:22:34 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.366 (sec). Leaf size: 53

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

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

✓ Solution by Mathematica

Time used: 0.135 (sec). Leaf size: 121

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

\[ y(x)\to (-1)^{-\frac {b}{n}} n^{\frac {b-n-1}{n}} a^{\frac {1-b}{n}} e^{-\frac {a x^n}{n}} \left (x^n\right )^{\frac {1-b}{n}} \left (-(b-1) c_1 (-1)^{\frac {1}{n}} \Gamma \left (\frac {b-1}{n},-\frac {a x^n}{n}\right )+c_2 n (-1)^{b/n}+(b-1) c_1 (-1)^{\frac {1}{n}} \operatorname {Gamma}\left (\frac {b-1}{n}\right )\right ) \]