59.1.6 problem 6

Internal problem ID [9178]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 6
Date solved : Monday, January 27, 2025 at 05:51:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 5 y^{\prime \prime }-2 x y^{\prime }+10 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.022 (sec). Leaf size: 31

dsolve(5*diff(y(x),x$2)-2*x*diff(y(x),x)+10*y(x)=0,y(x), singsol=all)
 

\[ y = c_{2} \operatorname {hypergeom}\left (\left [-\frac {5}{2}\right ], \left [\frac {1}{2}\right ], \frac {x^{2}}{5}\right )+\frac {4 x \left (x^{4}-25 x^{2}+\frac {375}{4}\right ) c_{1}}{375} \]

✓ Solution by Mathematica

Time used: 0.117 (sec). Leaf size: 138

DSolve[5*D[y[x],{x,2}]-2*x*D[y[x],x]+10*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {1}{200} \sqrt {\frac {\pi }{5}} c_2 \sqrt {x^2} \left (4 x^4-100 x^2+375\right ) \text {erfi}\left (\frac {\sqrt {x^2}}{\sqrt {5}}\right )+\frac {32 c_1 x^5}{25 \sqrt {5}}-\frac {32 c_1 x^3}{\sqrt {5}}-\frac {9}{20} c_2 e^{\frac {x^2}{5}} x^2+c_2 e^{\frac {x^2}{5}}+\frac {1}{50} c_2 e^{\frac {x^2}{5}} x^4+24 \sqrt {5} c_1 x \]