49.19.9 problem 3(f)

Internal problem ID [7729]
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(f)
Date solved : Sunday, March 30, 2025 at 12:20:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 51

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

\[ y = c_1 \,x^{2} \left (1+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_2 \left (\ln \left (x \right ) \left (\left (-48\right ) x^{3}+\operatorname {O}\left (x^{8}\right )\right )+\left (12+36 x +72 x^{2}+88 x^{3}-24 x^{4}-\frac {24}{5} x^{5}-\frac {16}{15} x^{6}-\frac {8}{35} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x} \]

✓ Mathematica. Time used: 0.104 (sec). Leaf size: 58

ode=x^2*D[y[x],{x,2}]-2*x^2*D[y[x],x]+(4*x-2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 x^2+c_1 \left (-4 x^2 \log (x)-\frac {4 x^6+18 x^5+90 x^4-390 x^3-270 x^2-135 x-45}{45 x}\right ) \]

✓ Sympy. Time used: 0.863 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = C_{1} x^{2} + O\left (x^{8}\right ) \]