43.19.7 problem 3(d)

Internal problem ID [9013]
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(d)
Date solved : Tuesday, September 30, 2025 at 06:01:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 53

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

\[ y = c_1 \,x^{2} \left (1+x +\frac {2}{3} x^{2}+\frac {1}{3} x^{3}+\frac {2}{15} x^{4}+\frac {2}{45} x^{5}+\frac {4}{315} x^{6}+\frac {1}{315} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 x \left (1+2 x +2 x^{2}+\frac {4}{3} x^{3}+\frac {2}{3} x^{4}+\frac {4}{15} x^{5}+\frac {4}{45} x^{6}+\frac {8}{315} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 92

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

\[ y(x)\to c_1 \left (\frac {4 x^7}{45}+\frac {4 x^6}{15}+\frac {2 x^5}{3}+\frac {4 x^4}{3}+2 x^3+2 x^2+x\right )+c_2 \left (\frac {4 x^8}{315}+\frac {2 x^7}{45}+\frac {2 x^6}{15}+\frac {x^5}{3}+\frac {2 x^4}{3}+x^3+x^2\right ) \]

✓ Sympy. Time used: 0.325 (sec). Leaf size: 42

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

\[ y{\left (x \right )} = C_{2} x^{2} \left (\frac {2 x^{5}}{45} + \frac {2 x^{4}}{15} + \frac {x^{3}}{3} + \frac {2 x^{2}}{3} + x + 1\right ) + C_{1} x + O\left (x^{8}\right ) \]