73.23.24 problem 33.5 (L)

Internal problem ID [15599]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 33. Power series solutions I: Basic computational methods. Additional Exercises. page 641
Problem number : 33.5 (L)
Date solved : Monday, March 31, 2025 at 01:42:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 44

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

\[ y = \left (1+\frac {1}{3} x^{3}+\frac {1}{20} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{6} x^{4}+\frac {1}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 49

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

\[ y(x)\to c_1 \left (\frac {x^5}{20}+\frac {x^3}{3}+1\right )+c_2 \left (\frac {x^5}{40}+\frac {x^4}{6}+\frac {x^3}{6}+x\right ) \]

✓ Sympy. Time used: 0.831 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{5}}{20} + \frac {x^{3}}{3} + 1\right ) + C_{1} x \left (\frac {x^{4}}{40} + \frac {x^{3}}{6} + \frac {x^{2}}{6} + 1\right ) + O\left (x^{6}\right ) \]