68.1.7 problem Problem 1.6(a)

Internal problem ID [14136]
Book : Differential Equations, Linear, Nonlinear, Ordinary, Partial. A.C. King, J.Billingham, S.R.Otto. Cambridge Univ. Press 2003
Section : Chapter 1 VARIABLE COEFFICIENT, SECOND ORDER DIFFERENTIAL EQUATIONS. Problems page 28
Problem number : Problem 1.6(a)
Date solved : Tuesday, January 28, 2025 at 06:15:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.058 (sec). Leaf size: 39

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

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

✓ Solution by Mathematica

Time used: 0.111 (sec). Leaf size: 37

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

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