Internal problem ID [6513]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 42.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {2 y^{\prime \prime } x^{2}+2 y^{\prime } x -x y-\cos \relax (x ) \sin \relax (x )=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 77
Order:=6; dsolve(2*x^2*diff(y(x), x, x) + 2*x*diff(y(x), x) - x*y(x) = cos(x)*sin(x),y(x),type='series',x=0);
\[ y \relax (x ) = \left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+\frac {1}{2} x +\frac {1}{16} x^{2}+\frac {1}{288} x^{3}+\frac {1}{9216} x^{4}+\frac {1}{460800} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+x \left (\frac {1}{2}+\frac {1}{16} x -\frac {29}{864} x^{2}-\frac {29}{27648} x^{3}+\frac {18287}{6912000} x^{4}+\frac {18287}{497664000} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (-x -\frac {3}{16} x^{2}-\frac {11}{864} x^{3}-\frac {25}{55296} x^{4}-\frac {137}{13824000} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.254 (sec). Leaf size: 340
AsymptoticDSolveValue[2*x^2*y''[x]+2*x*y'[x]-x*y[x]==Cos[x]*Sin[x],y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^5}{460800}+\frac {x^4}{9216}+\frac {x^3}{288}+\frac {x^2}{16}+\frac {x}{2}+1\right )+c_1 \left (x^5 \left (\frac {\log (x)}{460800}-\frac {107}{13824000}\right )+x^4 \left (\frac {\log (x)}{9216}-\frac {19}{55296}\right )+x^3 \left (\frac {\log (x)}{288}-\frac {1}{108}\right )+x^2 \left (\frac {\log (x)}{16}-\frac {1}{8}\right )+x \left (\frac {\log (x)}{2}-\frac {1}{2}\right )+\log (x)+1\right )+\left (\frac {88963 x^6}{16588800}+\frac {4229 x^5}{460800}-\frac {95 x^4}{2304}-\frac {29 x^3}{288}+\frac {x^2}{8}+\frac {x}{2}\right ) \left (x^5 \left (\frac {\log (x)}{460800}-\frac {107}{13824000}\right )+x^4 \left (\frac {\log (x)}{9216}-\frac {19}{55296}\right )+x^3 \left (\frac {\log (x)}{288}-\frac {1}{108}\right )+x^2 \left (\frac {\log (x)}{16}-\frac {1}{8}\right )+x \left (\frac {\log (x)}{2}-\frac {1}{2}\right )+\log (x)+1\right )+\left (\frac {x^5}{460800}+\frac {x^4}{9216}+\frac {x^3}{288}+\frac {x^2}{16}+\frac {x}{2}+1\right ) \left (\frac {x^6 (1476968-1334445 \log (x))}{248832000}+\frac {x^5 (-126870 \log (x)-273671)}{13824000}+\frac {5 x^4 (228 \log (x)-281)}{27648}+\frac {1}{864} x^3 (87 \log (x)+85)+\frac {1}{16} x^2 (3-2 \log (x))-\frac {1}{2} x \log (x)\right ) \]