12.12.7 problem 7

Internal problem ID [1861]
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 : 7
Date solved : Monday, January 27, 2025 at 05:37:26 AM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-5 x y^{\prime }-4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 39

Order:=6; 
dsolve((1-x^2)*diff(y(x),x$2)-5*x*diff(y(x),x)-4*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1+2 x^{2}+\frac {8}{3} x^{4}\right ) y \left (0\right )+\left (x +\frac {3}{2} x^{3}+\frac {15}{8} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

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

AsymptoticDSolveValue[(1-x^2)*D[y[x],{x,2}]-5*x*D[y[x],x]-4*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {15 x^5}{8}+\frac {3 x^3}{2}+x\right )+c_1 \left (\frac {8 x^4}{3}+2 x^2+1\right ) \]