61.28.36 problem 96

Internal problem ID [12596]
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 : 96
Date solved : Tuesday, January 28, 2025 at 03:22:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.372 (sec). Leaf size: 56

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

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

✓ Solution by Mathematica

Time used: 0.093 (sec). Leaf size: 93

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

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