Internal problem ID [12078]
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]]
\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (1-\frac {1}{4 x^{2}}\right ) y=x} \]
✓ 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: 0.443 (sec). Leaf size: 111
DSolve[y''[x]+1/x*y'[x]+(1-1/(4*x^2))*y[x]==x,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-i x} \left (-\frac {e^{2 i x} x^{3/2} \Gamma \left (\frac {5}{2},i x\right )}{\sqrt {-i x}}+\sqrt {x^2} \left (2 c_1-i c_2 e^{2 i x}\right )+\frac {(i x)^{3/2} \Gamma \left (\frac {5}{2},-i x\right )}{\sqrt {x}}\right )}{2 \sqrt {x} \sqrt {x^2}} \]