61.28.32 problem 92

Internal problem ID [12592]
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 : 92
Date solved : Tuesday, January 28, 2025 at 03:22:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.770 (sec). Leaf size: 122

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

\[ y = \frac {c_{2} {\mathrm e}^{-\frac {a \,x^{n}}{2 n}} n \left (\left (n +1\right ) x^{-\frac {3 n}{2}+\frac {1}{2}}+a \,x^{-\frac {n}{2}+\frac {1}{2}}\right ) \operatorname {WhittakerM}\left (-\frac {n -1}{2 n}, \frac {2 n +1}{2 n}, \frac {a \,x^{n}}{n}\right )+c_{2} x^{-\frac {3 n}{2}+\frac {1}{2}} {\mathrm e}^{-\frac {a \,x^{n}}{2 n}} \left (n +1\right )^{2} \operatorname {WhittakerM}\left (\frac {n +1}{2 n}, \frac {2 n +1}{2 n}, \frac {a \,x^{n}}{n}\right )+c_{1}}{x} \]

✓ Solution by Mathematica

Time used: 0.058 (sec). Leaf size: 62

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

\[ y(x)\to (-1)^{-1/n} n^{\frac {1}{n}-1} a^{-1/n} \left (x^n\right )^{-1/n} \left (c_1 (-1)^{\frac {1}{n}} \Gamma \left (\frac {1}{n},0,\frac {a x^n}{n}\right )+c_2 n\right ) \]