46.1.12 problem 18

Internal problem ID [7302]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.1. page 174
Problem number : 18
Date solved : Monday, January 27, 2025 at 02:49:57 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+30 y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&={\frac {15}{8}} \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 14

Order:=6; 
dsolve([(1-x^2)*diff(y(x),x$2)-2*x*diff(y(x),x)+30*y(x)=0,y(0) = 0, D(y)(0) = 15/8],y(x),type='series',x=0);
 

\[ y = \frac {15}{8} x -\frac {35}{4} x^{3}+\frac {63}{8} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 23

AsymptoticDSolveValue[{(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+30*y[x]==0,{y[0]==0,Derivative[1][y][0] ==1875/1000}},y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {63 x^5}{8}-\frac {35 x^3}{4}+\frac {15 x}{8} \]