43.19.6 problem 3(c)

Internal problem ID [9012]
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 : 3(c)
Date solved : Tuesday, September 30, 2025 at 06:01:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+5 x y^{\prime }+\left (-x^{3}+3\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 35

Order:=8; 
ode:=x^2*diff(diff(y(x),x),x)+5*x*diff(y(x),x)+(-x^3+3)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1+\frac {1}{15} x^{3}+\frac {1}{720} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{x}+\frac {c_2 \left (-2-\frac {2}{3} x^{3}-\frac {1}{36} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 40

ode=x^2*D[y[x],{x,2}]+5*x*D[y[x],x]+(3-3*x^3)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {x^3}{8}+\frac {1}{x^3}+1\right )+c_2 \left (\frac {x^5}{80}+\frac {x^2}{5}+\frac {1}{x}\right ) \]

✓ Sympy. Time used: 0.332 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + 5*x*Derivative(y(x), x) + (3 - x**3)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = \frac {C_{2} \left (\frac {x^{6}}{720} + \frac {x^{3}}{15} + 1\right )}{x} + \frac {C_{1} \left (\frac {x^{9}}{4536} + \frac {x^{6}}{72} + \frac {x^{3}}{3} + 1\right )}{x^{3}} + O\left (x^{8}\right ) \]