59.1.87 problem 89

Internal problem ID [9259]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 89
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 (2 x +3\right ) y^{\prime }-\left (1-x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 1.849 (sec). Leaf size: 52

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

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

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 33

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

\[ y(x)\to \frac {e^{-x} \left (c_2 x^{3/2} L_{-\frac {3}{2}}^{\frac {3}{2}}(x)+c_1\right )}{x} \]