Internal problem ID [6504]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 33.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {4 x y^{\prime \prime }+2 y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 44
Order:=6; dsolve(4*x*diff(y(x), x$2) +2*diff(y(x),x)+y(x) = 0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1-\frac {1}{6} x +\frac {1}{120} x^{2}-\frac {1}{5040} x^{3}+\frac {1}{362880} x^{4}-\frac {1}{39916800} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (1-\frac {1}{2} x +\frac {1}{24} x^{2}-\frac {1}{720} x^{3}+\frac {1}{40320} x^{4}-\frac {1}{3628800} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 85
AsymptoticDSolveValue[4*x*y''[x] +2*y'[x]+y[x] == 0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (-\frac {x^5}{39916800}+\frac {x^4}{362880}-\frac {x^3}{5040}+\frac {x^2}{120}-\frac {x}{6}+1\right )+c_2 \left (-\frac {x^5}{3628800}+\frac {x^4}{40320}-\frac {x^3}{720}+\frac {x^2}{24}-\frac {x}{2}+1\right ) \]