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]]
Solve \begin {gather*} \boxed {2 x y^{\prime \prime }+2 y^{\prime }+2 y-\sin \left (\sqrt {x}\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (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)
\[ y \relax (x ) = \BesselJ \left (0, 2 \sqrt {x}\right ) c_{2}+\BesselY \left (0, 2 \sqrt {x}\right ) c_{1}-\frac {\pi \left (\BesselJ \left (0, 2 \sqrt {x}\right ) \left (\int \BesselY \left (0, 2 \sqrt {x}\right ) \sin \left (\sqrt {x}\right )d x \right )-\BesselY \left (0, 2 \sqrt {x}\right ) \left (\int \BesselJ \left (0, 2 \sqrt {x}\right ) \sin \left (\sqrt {x}\right )d x \right )\right )}{2} \]
✓ Solution by Mathematica
Time used: 16.105 (sec). Leaf size: 88
DSolve[2*x*y''[x]+2*y'[x]+2*y[x]==Sin[Sqrt[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \, _0\tilde {F}_1(;1;-x) \int _1^x-\frac {1}{2} \pi Y_0\left (2 \sqrt {K[1]}\right ) \sin \left (\sqrt {K[1]}\right )dK[1]+2 Y_0\left (2 \sqrt {x}\right ) \left (\int _1^x\frac {1}{4} \pi \, _0\tilde {F}_1(;1;-K[2]) \sin \left (\sqrt {K[2]}\right )dK[2]+c_2\right )+c_1 \, _0\tilde {F}_1(;1;-x) \\ \end{align*}