59.1.88 problem 90

Internal problem ID [9260]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 90
Date solved : Monday, January 27, 2025 at 06:00:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 52

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

\[ y = \frac {i {\mathrm e}^{-\frac {x}{2}} x^{{5}/{2}} \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, \sqrt {x}}{2}\right ) \sqrt {\pi }\, c_{2} +c_{1} x^{{5}/{2}} {\mathrm e}^{-\frac {x}{2}}+2 c_{2} \left (x^{2}+x +3\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.381 (sec). Leaf size: 94

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

\[ y(x)\to \frac {e^{-\frac {x}{2}-\frac {5}{4}} \left (15 c_1 x^{5/2}-2 c_2 e^{x/2} x^2-6 c_2 e^{x/2}-2 c_2 e^{x/2} x+\sqrt {2} c_2 (-x)^{5/2} \Gamma \left (\frac {1}{2},-\frac {x}{2}\right )\right )}{15 x^2} \]