59.1.529 problem 545

Internal problem ID [9701]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 545
Date solved : Monday, January 27, 2025 at 06:13:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} y^{\prime \prime }+3 x \left (x^{2}+3\right ) y^{\prime }-\left (-5 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 37

dsolve(9*x^2*diff(y(x),x$2)+3*x*(3+x^2)*diff(y(x),x)-(1-5*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-\frac {x^{2}}{12}} \left ({\mathrm e}^{-\frac {x^{2}}{12}} c_{2} x +\operatorname {WhittakerM}\left (\frac {1}{3}, \frac {1}{6}, \frac {x^{2}}{6}\right ) x^{{1}/{3}} c_{1} \right )}{x^{{4}/{3}}} \]

✓ Solution by Mathematica

Time used: 0.115 (sec). Leaf size: 70

DSolve[9*x^2*D[y[x],{x,2}]+3*x*(3+x^2)*D[y[x],x]-(1-5*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {e^{-\frac {x^2}{6}-\frac {1}{6}} \left (2 \sqrt [3]{e} c_1 x^{4/3}+\sqrt [3]{6} c_2 \left (-x^2\right )^{2/3} \Gamma \left (\frac {1}{3},-\frac {x^2}{6}\right )\right )}{2 x^{5/3}} \]