88.24.6 problem 6

Internal problem ID [24199]
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 : 6
Date solved : Thursday, October 02, 2025 at 10:00:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 \left (-x^{2}+1\right ) y^{\prime \prime }-8 x y^{\prime }+3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 39

Order:=6; 
ode:=4*(-x^2+1)*diff(diff(y(x),x),x)-8*x*diff(y(x),x)+3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {3}{8} x^{2}-\frac {21}{128} x^{4}\right ) y \left (0\right )+\left (x +\frac {5}{24} x^{3}+\frac {15}{128} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 42

ode=4*(1-x^2)*D[y[x],{x,2}]-8*x*D[y[x],x]+3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {15 x^5}{128}+\frac {5 x^3}{24}+x\right )+c_1 \left (-\frac {21 x^4}{128}-\frac {3 x^2}{8}+1\right ) \]

✓ Sympy. Time used: 0.255 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*x*Derivative(y(x), x) + (4 - 4*x**2)*Derivative(y(x), (x, 2)) + 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {21 x^{4}}{128} - \frac {3 x^{2}}{8} + 1\right ) + C_{1} x \left (\frac {5 x^{2}}{24} + 1\right ) + O\left (x^{6}\right ) \]