Internal problem ID [6520]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 49.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x^{2}+\left (x^{2}+6 x \right ) y^{\prime }+x y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 44
Order:=6; dsolve(x^2*diff(y(x), x$2)+(6*x+x^2)*diff(y(x), x)+x*y(x) = 0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \left (1-\frac {1}{6} x +\frac {1}{42} x^{2}-\frac {1}{336} x^{3}+\frac {1}{3024} x^{4}-\frac {1}{30240} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (2880-2880 x +1440 x^{2}-480 x^{3}+120 x^{4}-24 x^{5}+\mathrm {O}\left (x^{6}\right )\right )}{x^{5}} \]
✓ Solution by Mathematica
Time used: 0.042 (sec). Leaf size: 68
AsymptoticDSolveValue[x^2*y''[x]+(6*x+x^2)*y'[x]+x*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^4}{3024}-\frac {x^3}{336}+\frac {x^2}{42}-\frac {x}{6}+1\right )+c_1 \left (\frac {1}{x^5}-\frac {1}{x^4}+\frac {1}{2 x^3}-\frac {1}{6 x^2}+\frac {1}{24 x}\right ) \]