7.16.2 problem 2

Internal problem ID [499]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.4 (Method of Frobenius: The exceptional cases). Problems at page 246
Problem number : 2
Date solved : Tuesday, March 04, 2025 at 11:25:26 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x y^{\prime \prime }+\left (5-x \right ) y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 44

Order:=6; 
ode:=x*diff(diff(y(x),x),x)+(5-x)*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1+\frac {1}{5} x +\frac {1}{30} x^{2}+\frac {1}{210} x^{3}+\frac {1}{1680} x^{4}+\frac {1}{15120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-144-144 x -72 x^{2}-24 x^{3}-6 x^{4}-\frac {6}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{4}} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 62

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

\[ y(x)\to c_1 \left (\frac {1}{x^4}+\frac {1}{x^3}+\frac {1}{2 x^2}+\frac {1}{6 x}+\frac {1}{24}\right )+c_2 \left (\frac {x^4}{1680}+\frac {x^3}{210}+\frac {x^2}{30}+\frac {x}{5}+1\right ) \]

✓ Sympy. Time used: 0.903 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (5 - x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{5}}{15120} + \frac {x^{4}}{1680} + \frac {x^{3}}{210} + \frac {x^{2}}{30} + \frac {x}{5} + 1\right ) + \frac {C_{1} \left (\frac {x^{3}}{6} + \frac {x^{2}}{2} + x + 1\right )}{x^{4}} + O\left (x^{6}\right ) \]