62.1.7 problem Problem 1.6(a)

Internal problem ID [15417]
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 : Thursday, October 02, 2025 at 10:13:54 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*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 39

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

\[ y = \frac {{\mathrm e}^{-x} \left (\operatorname {erf}\left (\sqrt {-x}\right ) \sqrt {\pi }\, c_1 x +2 c_2 \sqrt {x}\, \sqrt {-x}\right )}{2 \sqrt {-x}} \]

✓ Mathematica. Time used: 0.067 (sec). Leaf size: 37

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

\begin{align*} y(x)&\to e^{-x} \left (c_2 \sqrt {x}+c_1 \sqrt {-x} \Gamma \left (\frac {1}{2},-x\right )\right ) \end{align*}

✓ Sympy. Time used: 0.856 (sec). Leaf size: 542

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(x - 1/2)*Derivative(y(x), x) + y(x)/2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \text {Solution too large to show} \]