43.19.5 problem 3(b)

Internal problem ID [9011]
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(b)
Date solved : Tuesday, September 30, 2025 at 06:01:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }-2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 53

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

\[ y = c_1 \,x^{2} \left (1-x +\frac {3}{5} x^{2}-\frac {4}{15} x^{3}+\frac {2}{21} x^{4}-\frac {1}{35} x^{5}+\frac {1}{135} x^{6}-\frac {8}{4725} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_2 \left (12-12 x +8 x^{3}-8 x^{4}+\frac {24}{5} x^{5}-\frac {32}{15} x^{6}+\frac {16}{21} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 87

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

\[ y(x)\to c_1 \left (-\frac {8 x^5}{45}+\frac {2 x^4}{5}-\frac {2 x^3}{3}+\frac {2 x^2}{3}+\frac {1}{x}-1\right )+c_2 \left (\frac {x^8}{135}-\frac {x^7}{35}+\frac {2 x^6}{21}-\frac {4 x^5}{15}+\frac {3 x^4}{5}-x^3+x^2\right ) \]

✓ Sympy. Time used: 0.275 (sec). Leaf size: 46

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)) - 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} x^{2} \left (- \frac {x^{5}}{35} + \frac {2 x^{4}}{21} - \frac {4 x^{3}}{15} + \frac {3 x^{2}}{5} - x + 1\right ) + \frac {C_{1} \left (x - 1\right )}{x} + O\left (x^{8}\right ) \]