Internal problem ID [2435]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition,
2015
Section: Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page
771
Problem number: (e).
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 {x^{2} y^{\prime \prime }+y^{\prime } x^{2}+y x=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 58
Order:=6; dsolve(x^2*diff(y(x),x$2)+x^2*diff(y(x),x)+x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (-x +x^{2}-\frac {1}{2} x^{3}+\frac {1}{6} x^{4}-\frac {1}{24} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (1-x +\frac {1}{4} x^{3}-\frac {5}{36} x^{4}+\frac {13}{288} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.02 (sec). Leaf size: 80
AsymptoticDSolveValue[x^2*y''[x]+x^2*y'[x]+x*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {1}{6} x \left (x^3-3 x^2+6 x-6\right ) \log (x)+\frac {1}{36} \left (-11 x^4+27 x^3-36 x^2+36\right )\right )+c_2 \left (\frac {x^5}{24}-\frac {x^4}{6}+\frac {x^3}{2}-x^2+x\right ) \]