61.29.36 problem 145

Internal problem ID [12645]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 145
Date solved : Tuesday, January 28, 2025 at 03:24:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.326 (sec). Leaf size: 141

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

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

✓ Solution by Mathematica

Time used: 0.090 (sec). Leaf size: 76

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

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