10.13.13 problem 16

Internal problem ID [1373]
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 : 16
Date solved : Monday, January 27, 2025 at 04:56:19 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*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 20

Order:=6; 
dsolve([(2+x^2)*diff(y(x),x$2)-x*diff(y(x),x)+4*y(x)=0,y(0) = -1, D(y)(0) = 3],y(x),type='series',x=0);
 

\[ y = -1+3 x +x^{2}-\frac {3}{4} x^{3}-\frac {1}{6} x^{4}+\frac {21}{160} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 32

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

\[ y(x)\to \frac {21 x^5}{160}-\frac {x^4}{6}-\frac {3 x^3}{4}+x^2+3 x-1 \]