61.27.35 problem 45

Internal problem ID [12545]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 45
Date solved : Tuesday, January 28, 2025 at 08:02:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 1.585 (sec). Leaf size: 96

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

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

✓ Solution by Mathematica

Time used: 0.091 (sec). Leaf size: 120

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

\[ y(x)\to c_2 \left (\frac {1}{n}+1\right )^{-\frac {1}{n+1}} n^{-\frac {1}{n+1}} a^{\frac {1}{n+1}} \left (x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric1F1}\left (\frac {a+b}{n a+a},\frac {n+2}{n+1},-\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {b}{n a+a},\frac {n}{n+1},-\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right ) \]