88.24.5 problem 5

Internal problem ID [24198]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 7. Series Methods. Exercises at page 202
Problem number : 5
Date solved : Thursday, October 02, 2025 at 10:00:45 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+56 y&=0 \end{align*}

Using series method with expansion around

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

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

Order:=6; 
ode:=(-x^2+1)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+56*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-28 x^{2}+\frac {350}{3} x^{4}\right ) y \left (0\right )+\left (x -9 x^{3}+\frac {99}{5} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 38

ode=(1-x^2)*D[y[x],{x,2}]-2*x*D[y[x],x]+56*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {99 x^5}{5}-9 x^3+x\right )+c_1 \left (\frac {350 x^4}{3}-28 x^2+1\right ) \]

✓ Sympy. Time used: 0.256 (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)) + 56*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {350 x^{4}}{3} - 28 x^{2} + 1\right ) + C_{1} x \left (1 - 9 x^{2}\right ) + O\left (x^{6}\right ) \]