29.9.15 problem 8, using series method

Internal problem ID [7385]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 12, Series Solutions of Differential Equations. Section 1. Miscellaneous problems. page 564
Problem number : 8, using series method
Date solved : Tuesday, September 30, 2025 at 04:30:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 28

Order:=6; 
ode:=(x^2+2*x)*diff(diff(y(x),x),x)-2*(1+x)*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{2} \left (1+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (-2-2 x -\frac {1}{2} x^{2}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 23

ode=(x^2+2*x)*D[y[x],{x,2}]-2*(x+1)*D[y[x],x]+2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 x^2+c_1 \left (\frac {x^2}{4}+x+1\right ) \]

✓ Sympy. Time used: 0.332 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-2*x - 2)*Derivative(y(x), x) + (x**2 + 2*x)*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=6)
 

\[ y{\left (x \right )} = C_{1} x^{2} \left (- \frac {x^{3}}{45} + \frac {x^{2}}{6} - \frac {2 x}{3} + 1\right ) + O\left (x^{6}\right ) \]