Internal problem ID [11052]
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]]
\[ \boxed {x^{2} y^{\prime \prime }+x \left (x -\frac {1}{2}\right ) y^{\prime }+\frac {y}{2}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 27
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)
\[ y \left (x \right ) = c_{1} \operatorname {WhittakerM}\left (\frac {1}{4}, \frac {1}{4}, x\right ) x^{\frac {1}{4}} {\mathrm e}^{-\frac {x}{2}}+c_{2} \sqrt {x}\, {\mathrm e}^{-x} \]
✓ Solution by Mathematica
Time used: 0.048 (sec). Leaf size: 32
DSolve[x^2*y''[x]+x*(x-1/2)*y'[x]+1/2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-x} \left (c_2 \sqrt {x}-c_1 x \operatorname {ExpIntegralE}\left (\frac {1}{2},-x\right )\right ) \\ \end{align*}