Internal problem ID [11050]
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 1.3(c).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (1-\frac {1}{4 x^{2}}\right ) y-x=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 71
dsolve(diff(y(x),x$2)+1/x*diff(y(x),x)+(1-1/(4*x^2))*y(x)=x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{2} \sin \left (x \right )}{\sqrt {x}}+\frac {\cos \left (x \right ) c_{1}}{\sqrt {x}}-\frac {3 \left (\sin \left (x \right ) \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right )-\cos \left (x \right ) \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right )-\frac {4 x^{\frac {3}{2}}}{3}\right )}{4 \sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.134 (sec). Leaf size: 74
DSolve[y''[x]+1/x*y'[x]+(1-1/(4*x^2))*y[x]==x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {i e^{-i x} \left (x^{5/2} \left (-\operatorname {ExpIntegralE}\left (-\frac {3}{2},-i x\right )\right )+e^{2 i x} \left (x^{5/2} \operatorname {ExpIntegralE}\left (-\frac {3}{2},i x\right )-c_2\right )-2 i c_1\right )}{2 \sqrt {x}} \\ \end{align*}