68.1.5 problem Problem 1.3(c)

Internal problem ID [14134]
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)
Date solved : Tuesday, January 28, 2025 at 06:15:33 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{x}+\left (1-\frac {1}{4 x^{2}}\right ) y&=x \end{align*}

✓ Solution by Maple

Time used: 0.160 (sec). Leaf size: 62

dsolve(diff(y(x),x$2)+1/x*diff(y(x),x)+(1-1/(4*x^2))*y(x)=x,y(x), singsol=all)
 

\[ y = \frac {\sin \left (x \right ) c_{2} +\cos \left (x \right ) c_{1} -\frac {3 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right ) \sin \left (x \right )}{4}+\frac {3 \sqrt {2}\, \cos \left (x \right ) \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right )}{4}+x^{{3}/{2}}}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.320 (sec). Leaf size: 111

DSolve[D[y[x],{x,2}]+1/x*D[y[x],x]+(1-1/(4*x^2))*y[x]==x,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {e^{-i x} \left (-\frac {e^{2 i x} x^{3/2} \Gamma \left (\frac {5}{2},i x\right )}{\sqrt {-i x}}+\sqrt {x^2} \left (2 c_1-i c_2 e^{2 i x}\right )+\frac {(i x)^{3/2} \Gamma \left (\frac {5}{2},-i x\right )}{\sqrt {x}}\right )}{2 \sqrt {x} \sqrt {x^2}} \]