43.19.3 problem 1(iii)

Internal problem ID [9009]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 166
Problem number : 1(iii)
Date solved : Tuesday, September 30, 2025 at 06:01:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 56

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

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+3 x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{16} x^{4}-\frac {43}{1200} x^{5}+\frac {161}{7200} x^{6}-\frac {1837}{117600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\left (-9\right ) x -\frac {7}{2} x^{2}+\frac {7}{9} x^{3}-\frac {25}{96} x^{4}+\frac {5141}{36000} x^{5}-\frac {2083}{24000} x^{6}+\frac {489941}{8232000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2}{x} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 84

ode=(1-x^2)*D[y[x],{x,2}]+3*(x+x^2)*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (\frac {53 x^7}{630}+\frac {5 x^6}{24}+\frac {2 x^5}{15}-\frac {x^4}{4}-\frac {2 x^3}{3}+x\right )+c_1 \left (-\frac {19 x^7}{420}-\frac {x^6}{144}+\frac {3 x^5}{20}+\frac {5 x^4}{24}-\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.372 (sec). Leaf size: 8

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - x**2)*Derivative(y(x), (x, 2)) + (3*x**2 + 3*x)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = \frac {C_{1}}{x} + O\left (x^{8}\right ) \]