61.27.41 problem 51

Internal problem ID [12551]
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 : 51
Date solved : Tuesday, January 28, 2025 at 08:02:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.832 (sec). Leaf size: 167

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

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

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

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

Not solved