10.14.10 problem 7. case \(x_0=0\)

Internal problem ID [1392]
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 \(x_0=0\)
Date solved : Monday, January 27, 2025 at 04:56:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{3}+1\right ) y^{\prime \prime }+4 x y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 44

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);
 

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {3}{8} x^{4}+\frac {1}{20} x^{5}\right ) y \left (0\right )+\left (x -\frac {5}{6} x^{3}+\frac {13}{24} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 49

AsymptoticDSolveValue[(1+x^3)*D[y[x],{x,2}]+4*x*D[y[x],x]+y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {13 x^5}{24}-\frac {5 x^3}{6}+x\right )+c_1 \left (\frac {x^5}{20}+\frac {3 x^4}{8}-\frac {x^2}{2}+1\right ) \]