7.16.5 problem 5

Internal problem ID [502]
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 : 5
Date solved : Thursday, March 13, 2025 at 03:36:05 PM
CAS classification : [_Laguerre]

\begin{align*} x y^{\prime \prime }-\left (4+x \right ) y^{\prime }+3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 42

Order:=6; 
ode:=x*diff(diff(y(x),x),x)-(x+4)*diff(y(x),x)+3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{5} \left (1+\frac {1}{3} x +\frac {1}{14} x^{2}+\frac {1}{84} x^{3}+\frac {5}{3024} x^{4}+\frac {1}{5040} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (2880+2160 x +720 x^{2}+120 x^{3}-6 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 63

ode=x*D[y[x],{x,2}]-(4+x)*D[y[x],x]+3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^3}{24}+\frac {x^2}{4}+\frac {3 x}{4}+1\right )+c_2 \left (\frac {5 x^9}{3024}+\frac {x^8}{84}+\frac {x^7}{14}+\frac {x^6}{3}+x^5\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) - (x + 4)*Derivative(y(x), x) + 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : Expected Expr or iterable but got None