42.1.6 problem 3.24 (a)

Internal problem ID [6828]
Book : Advanced Mathematical Methods for Scientists and Engineers, Bender and Orszag. Springer October 29, 1999
Section : Chapter 3. APPROXIMATE SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS. page 136
Problem number : 3.24 (a)
Date solved : Sunday, March 30, 2025 at 11:23:52 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (x +2\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: 0.031 (sec). Leaf size: 36

Order:=6; 
ode:=x*(x+2)*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);
 

\[ y = \left (-\frac {5}{2} x -\frac {3}{8} x^{2}+\frac {1}{12} x^{3}-\frac {5}{192} x^{4}+\frac {3}{320} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1+x +\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 53

ode=x*(x+2)*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}]
 

\[ y(x)\to c_2 \left (\frac {3 x^5}{320}-\frac {5 x^4}{192}+\frac {x^3}{12}-\frac {3 x^2}{8}-\frac {5 x}{2}+(x+1) \log (x)\right )+c_1 (x+1) \]

✓ Sympy. Time used: 1.176 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x + 2)*Derivative(y(x), (x, 2)) + (2*x + 2)*Derivative(y(x), x) - 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \left (\frac {x^{5}}{450} + \frac {x^{4}}{36} + \frac {2 x^{3}}{9} + x^{2} + 2 x + 1\right ) + O\left (x^{6}\right ) \]