Internal problem ID [12400]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime } x^{2}+x \left (x -\frac {1}{2}\right ) y^{\prime }+\frac {y}{2}=0} \]
✓ Solution by Maple
Time used: 0.032 (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)
\[ y \left (x \right ) = \frac {{\mathrm e}^{-x} \left (\operatorname {erf}\left (\sqrt {-x}\right ) \sqrt {\pi }\, c_{1} x +2 c_{2} \sqrt {x}\, \sqrt {-x}\right )}{2 \sqrt {-x}} \]
✓ Solution by Mathematica
Time used: 0.084 (sec). Leaf size: 37
DSolve[x^2*y''[x]+x*(x-1/2)*y'[x]+1/2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-x} \left (c_2 \sqrt {x}+c_1 \sqrt {-x} \Gamma \left (\frac {1}{2},-x\right )\right ) \]