Internal problem ID [5705]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 4. Power Series Solutions and Special Functions. Section 4.4. REGULAR SINGULAR
POINTS. Page 175
Problem number: 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Lienard]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+y x^{2}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.021 (sec). Leaf size: 47
Order:=8; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+x^2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{64} x^{4}-\frac {1}{2304} x^{6}+\mathrm {O}\left (x^{8}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\frac {11}{13824} x^{6}+\mathrm {O}\left (x^{8}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 81
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+x^2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {x^6}{2304}+\frac {x^4}{64}-\frac {x^2}{4}+1\right )+c_2 \left (\frac {11 x^6}{13824}-\frac {3 x^4}{128}+\frac {x^2}{4}+\left (-\frac {x^6}{2304}+\frac {x^4}{64}-\frac {x^2}{4}+1\right ) \log (x)\right ) \]