10.13.5 problem 6

Internal problem ID [1365]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 5.2, Series Solutions Near an Ordinary Point, Part I. page 263
Problem number : 6
Date solved : Monday, January 27, 2025 at 04:56:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+2\right ) y^{\prime \prime }-x y^{\prime }+4 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: 39

Order:=6; 
dsolve((2+x^2)*diff(y(x),x$2)-x*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1-x^{2}+\frac {1}{6} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{3}+\frac {7}{160} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 40

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

\[ y(x)\to c_2 \left (\frac {7 x^5}{160}-\frac {x^3}{4}+x\right )+c_1 \left (\frac {x^4}{6}-x^2+1\right ) \]