Internal problem ID [1168]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {2 x y^{\prime \prime }+2 y^{\prime }+2 y=\sin \left (\sqrt {x}\right )} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 72
dsolve(2*x*diff(y(x),x$2)+2*diff(y(x),x)+2*y(x)=sin(sqrt(x)),y(x), singsol=all)
\[ y \left (x \right ) = \operatorname {BesselJ}\left (0, 2 \sqrt {x}\right ) c_{2} +\operatorname {BesselY}\left (0, 2 \sqrt {x}\right ) c_{1} +\frac {\pi \left (\left (\int \operatorname {BesselJ}\left (0, 2 \sqrt {x}\right ) \sin \left (\sqrt {x}\right )d x \right ) \operatorname {BesselY}\left (0, 2 \sqrt {x}\right )-\left (\int \operatorname {BesselY}\left (0, 2 \sqrt {x}\right ) \sin \left (\sqrt {x}\right )d x \right ) \operatorname {BesselJ}\left (0, 2 \sqrt {x}\right )\right )}{2} \]
✓ Solution by Mathematica
Time used: 11.396 (sec). Leaf size: 110
DSolve[2*x*y''[x]+2*y'[x]+2*y[x]==Sin[Sqrt[x]],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \operatorname {BesselJ}\left (0,2 \sqrt {x}\right ) \int _1^x-\frac {1}{2} \pi \operatorname {BesselY}\left (0,2 \sqrt {K[1]}\right ) \sin \left (\sqrt {K[1]}\right )dK[1]+2 \operatorname {BesselY}\left (0,2 \sqrt {x}\right ) \int _1^x\frac {1}{4} \pi \operatorname {BesselJ}\left (0,2 \sqrt {K[2]}\right ) \sin \left (\sqrt {K[2]}\right )dK[2]+c_1 \operatorname {BesselJ}\left (0,2 \sqrt {x}\right )+2 c_2 \operatorname {BesselY}\left (0,2 \sqrt {x}\right ) \]