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 : Sunday, March 30, 2025 at 11:53:47 AM
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*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 14

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

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

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

ode=(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+30*y[x]==0; 
ic={y[0]==0,Derivative[1][y][0] ==1875/1000}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.899 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x*Derivative(y(x), x) + (1 - x**2)*Derivative(y(x), (x, 2)) + 30*y(x),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 15/8} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (30 x^{4} - 15 x^{2} + 1\right ) + C_{1} x \left (1 - \frac {14 x^{2}}{3}\right ) + O\left (x^{6}\right ) \]