59.1.530 problem 546

Internal problem ID [9702]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 546
Date solved : Monday, January 27, 2025 at 06:13:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 6 x^{2} y^{\prime \prime }+x \left (6 x^{2}+1\right ) y^{\prime }+\left (9 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.087 (sec). Leaf size: 36

dsolve(6*x^2*diff(y(x),x$2)+x*(1+6*x^2)*diff(y(x),x)+(1+9*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-\frac {x^{2}}{4}} \left ({\mathrm e}^{-\frac {x^{2}}{4}} x^{{11}/{12}} c_{2} +\operatorname {WhittakerM}\left (\frac {11}{24}, \frac {1}{24}, \frac {x^{2}}{2}\right ) c_{1} \right )}{x^{{7}/{12}}} \]

✓ Solution by Mathematica

Time used: 0.084 (sec). Leaf size: 61

DSolve[6*x^2*D[y[x],{x,2}]+x*(1+6*x^2)*D[y[x],x]+(1+9*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {e^{-\frac {x^2}{2}} \left (2 c_1 x^{11/6}+\sqrt [12]{2} c_2 \left (-x^2\right )^{11/12} \Gamma \left (\frac {1}{12},-\frac {x^2}{2}\right )\right )}{2 x^{3/2}} \]