Internal problem ID [4197]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter VII, Solutions in series. Examples XIV. page 177
Problem number: 11.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+y^{\prime } x^{2}+\left (-2+x \right ) y=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: 43
Order:=6; dsolve(x^2*diff(y(x),x$2)+x^2*diff(y(x),x)+(x-2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{2} \left (1-\frac {3}{4} x +\frac {3}{10} x^{2}-\frac {1}{12} x^{3}+\frac {1}{56} x^{4}-\frac {1}{320} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12-2 x^{3}+\frac {3}{2} x^{4}-\frac {3}{5} x^{5}+\mathrm {O}\left (x^{6}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.043 (sec). Leaf size: 60
AsymptoticDSolveValue[x^2*y''[x]+x^2*y'[x]+(x-2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {x^3}{8}-\frac {x^2}{6}+\frac {1}{x}\right )+c_2 \left (\frac {x^6}{56}-\frac {x^5}{12}+\frac {3 x^4}{10}-\frac {3 x^3}{4}+x^2\right ) \]