12.10.14 problem 14

Internal problem ID [1818]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.7 Variation of Parameters. Page 262
Problem number : 14
Date solved : Monday, January 27, 2025 at 05:36:04 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 2 x y^{\prime \prime }+2 y^{\prime }+2 y&=\sin \left (\sqrt {x}\right ) \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 71

dsolve(2*x*diff(y(x),x$2)+2*diff(y(x),x)+2*y(x)=sin(sqrt(x)),y(x), singsol=all)
 

\[ y = \operatorname {BesselJ}\left (0, 2 \sqrt {x}\right ) c_2 +\operatorname {BesselY}\left (0, 2 \sqrt {x}\right ) c_1 +\frac {\pi \left (\int \operatorname {BesselJ}\left (0, 2 \sqrt {x}\right ) \sin \left (\sqrt {x}\right )d x \right ) \operatorname {BesselY}\left (0, 2 \sqrt {x}\right )}{2}-\frac {\pi \left (\int \operatorname {BesselY}\left (0, 2 \sqrt {x}\right ) \sin \left (\sqrt {x}\right )d x \right ) \operatorname {BesselJ}\left (0, 2 \sqrt {x}\right )}{2} \]

✓ Solution by Mathematica

Time used: 11.671 (sec). Leaf size: 110

DSolve[2*x*D[y[x],{x,2}]+2*D[y[x],x]+2*y[x]==Sin[Sqrt[x]],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \operatorname {BesselJ}\left (0,2 \sqrt {x}\right ) \int _1^x-\frac {1}{2} \pi \operatorname {BesselY}\left (0,2 \sqrt {K[1]}\right ) \sin \left (\sqrt {K[1]}\right )dK[1]+2 \operatorname {BesselY}\left (0,2 \sqrt {x}\right ) \int _1^x\frac {1}{4} \pi \operatorname {BesselJ}\left (0,2 \sqrt {K[2]}\right ) \sin \left (\sqrt {K[2]}\right )dK[2]+c_1 \operatorname {BesselJ}\left (0,2 \sqrt {x}\right )+2 c_2 \operatorname {BesselY}\left (0,2 \sqrt {x}\right ) \]