43.17.1 problem 1(a)

Internal problem ID [8994]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 154
Problem number : 1(a)
Date solved : Tuesday, September 30, 2025 at 06:01:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\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: 0.022 (sec). Leaf size: 53

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

\[ y = c_1 x \left (1-\frac {1}{3} x +\frac {1}{12} x^{2}-\frac {1}{60} x^{3}+\frac {1}{360} x^{4}-\frac {1}{2520} x^{5}+\frac {1}{20160} x^{6}-\frac {1}{181440} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_2 \left (-2+2 x -x^{2}+\frac {1}{3} x^{3}-\frac {1}{12} x^{4}+\frac {1}{60} x^{5}-\frac {1}{360} x^{6}+\frac {1}{2520} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 92

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

\[ y(x)\to c_1 \left (\frac {x^5}{720}-\frac {x^4}{120}+\frac {x^3}{24}-\frac {x^2}{6}+\frac {x}{2}+\frac {1}{x}-1\right )+c_2 \left (\frac {x^7}{20160}-\frac {x^6}{2520}+\frac {x^5}{360}-\frac {x^4}{60}+\frac {x^3}{12}-\frac {x^2}{3}+x\right ) \]

✓ Sympy. Time used: 0.278 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (x**2 + 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)
 

\[ y{\left (x \right )} = C_{2} x \left (\frac {x^{6}}{20160} - \frac {x^{5}}{2520} + \frac {x^{4}}{360} - \frac {x^{3}}{60} + \frac {x^{2}}{12} - \frac {x}{3} + 1\right ) + \frac {C_{1} \left (1 - x\right )}{x} + O\left (x^{8}\right ) \]