42.1.10 problem 3.24 (e)

Internal problem ID [6832]
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 (e)
Date solved : Sunday, March 30, 2025 at 11:24:00 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 28

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

\[ y = c_1 \,x^{{3}/{2}} \left (1-\frac {1}{27} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-\frac {1}{9} x^{3}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 33

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

\[ y(x)\to c_2 \left (1-\frac {x^3}{9}\right )+c_1 \left (1-\frac {x^3}{27}\right ) x^{3/2} \]

✓ Sympy. Time used: 0.912 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = C_{2} \left (1 - \frac {x^{3}}{9}\right ) + C_{1} x^{\frac {3}{2}} \left (1 - \frac {x^{3}}{27}\right ) + O\left (x^{6}\right ) \]