Internal problem ID [2437]
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: 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} y^{\prime \prime }+2 y^{\prime } x^{2}+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 67
Order:=6; dsolve(4*x^2*diff(y(x),x$2)+2*x^2*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \sqrt {x}\, \left (\left (c_{2} \ln \relax (x )+c_{1}\right ) \left (1-\frac {1}{4} x +\frac {3}{64} x^{2}-\frac {5}{768} x^{3}+\frac {35}{49152} x^{4}-\frac {21}{327680} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (-\frac {1}{64} x^{2}+\frac {1}{256} x^{3}-\frac {19}{32768} x^{4}+\frac {25}{393216} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 129
AsymptoticDSolveValue[4*x^2*y''[x]+2*x^2*y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (-\frac {21 x^5}{327680}+\frac {35 x^4}{49152}-\frac {5 x^3}{768}+\frac {3 x^2}{64}-\frac {x}{4}+1\right )+c_2 \left (\sqrt {x} \left (\frac {25 x^5}{393216}-\frac {19 x^4}{32768}+\frac {x^3}{256}-\frac {x^2}{64}\right )+\sqrt {x} \left (-\frac {21 x^5}{327680}+\frac {35 x^4}{49152}-\frac {5 x^3}{768}+\frac {3 x^2}{64}-\frac {x}{4}+1\right ) \log (x)\right ) \]