Internal problem ID [6514]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 43.
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-x^{3}-x \sin \relax (x )=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 75
Order:=6; dsolve(2*x^2*diff(y(x), x, x) + 2*x*diff(y(x), x) - x*y(x) = x^3+x*sin(x),y(x),type='series',x=0);
\[ y \relax (x ) = \left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+\frac {1}{2} x +\frac {1}{16} x^{2}+\frac {1}{288} x^{3}+\frac {1}{9216} x^{4}+\frac {1}{460800} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{8}+\frac {1}{16} x -\frac {5}{1536} x^{2}-\frac {1}{15360} x^{3}+\mathrm {O}\left (x^{4}\right )\right )+\left (-x -\frac {3}{16} x^{2}-\frac {11}{864} x^{3}-\frac {25}{55296} x^{4}-\frac {137}{13824000} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.48 (sec). Leaf size: 268
AsymptoticDSolveValue[2*x^2*y''[x]+2*x*y'[x]-x*y[x]==x^3*x*Sin[x],y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^5}{460800}+\frac {x^4}{9216}+\frac {x^3}{288}+\frac {x^2}{16}+\frac {x}{2}+1\right )+c_1 \left (x^5 \left (\frac {\log (x)}{460800}-\frac {107}{13824000}\right )+x^4 \left (\frac {\log (x)}{9216}-\frac {19}{55296}\right )+x^3 \left (\frac {\log (x)}{288}-\frac {1}{108}\right )+x^2 \left (\frac {\log (x)}{16}-\frac {1}{8}\right )+x \left (\frac {\log (x)}{2}-\frac {1}{2}\right )+\log (x)+1\right )+\left (\frac {x^5}{460800}+\frac {x^4}{9216}+\frac {x^3}{288}+\frac {x^2}{16}+\frac {x}{2}+1\right ) \left (\frac {1}{144} x^6 (7-6 \log (x))+\frac {1}{50} x^5 (-5 \log (x)-4)\right )+\left (\frac {x^6}{24}+\frac {x^5}{10}\right ) \left (x^5 \left (\frac {\log (x)}{460800}-\frac {107}{13824000}\right )+x^4 \left (\frac {\log (x)}{9216}-\frac {19}{55296}\right )+x^3 \left (\frac {\log (x)}{288}-\frac {1}{108}\right )+x^2 \left (\frac {\log (x)}{16}-\frac {1}{8}\right )+x \left (\frac {\log (x)}{2}-\frac {1}{2}\right )+\log (x)+1\right ) \]