Internal problem ID [742]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Chapter 5.3, Series Solutions Near an Ordinary Point, Part II. page 269
Problem number: 7. case \(x_0=0\).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x^{3}+1\right ) y^{\prime \prime }+4 y^{\prime } x +y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 39
Order:=6; dsolve((1+x^3)*diff(y(x),x$2)+4*x*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {1}{2} x^{2}+\frac {3}{8} x^{4}+\frac {1}{20} x^{5}\right ) y \relax (0)+\left (x -\frac {5}{6} x^{3}+\frac {13}{24} x^{5}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 49
AsymptoticDSolveValue[(1+x^3)*y''[x]+4*x*y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {13 x^5}{24}-\frac {5 x^3}{6}+x\right )+c_1 \left (\frac {x^5}{20}+\frac {3 x^4}{8}-\frac {x^2}{2}+1\right ) \]