12.12.4 problem 4

Internal problem ID [1858]
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 : 4
Date solved : Monday, January 27, 2025 at 05:37:24 AM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }-12 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 39

Order:=6; 
dsolve((1-x^2)*diff(y(x),x$2)-8*x*diff(y(x),x)-12*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (15 x^{4}+6 x^{2}+1\right ) y \left (0\right )+\left (x +\frac {10}{3} x^{3}+7 x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 36

AsymptoticDSolveValue[(1-x^2)*D[y[x],{x,2}]-8*x*D[y[x],x]-12*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (7 x^5+\frac {10 x^3}{3}+x\right )+c_1 \left (15 x^4+6 x^2+1\right ) \]