12.12.19 problem 21

Internal problem ID [1873]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 21
Date solved : Monday, January 27, 2025 at 05:37:38 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (x^{2}-4\right ) y^{\prime \prime }-x y^{\prime }-3 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 )&=2 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 18

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

\[ y = -1+2 x +\frac {3}{8} x^{2}-\frac {1}{3} x^{3}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 29

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

\[ y(x)\to -\frac {3 x^4}{128}-\frac {x^3}{3}+\frac {3 x^2}{8}+2 x-1 \]