1.16.1 problem 1

Internal problem ID [498]
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 : 1
Date solved : Tuesday, September 30, 2025 at 03:59:32 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 44

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

\[ y = c_1 \left (1+\frac {1}{3} x +\frac {1}{12} x^{2}+\frac {1}{60} x^{3}+\frac {1}{360} x^{4}+\frac {1}{2520} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-2-2 x -x^{2}-\frac {1}{3} x^{3}-\frac {1}{12} x^{4}-\frac {1}{60} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 60

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

\[ y(x)\to c_1 \left (\frac {x^2}{24}+\frac {1}{x^2}+\frac {x}{6}+\frac {1}{x}+\frac {1}{2}\right )+c_2 \left (\frac {x^4}{360}+\frac {x^3}{60}+\frac {x^2}{12}+\frac {x}{3}+1\right ) \]

✓ Sympy. Time used: 0.287 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (3 - 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)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{5}}{2520} + \frac {x^{4}}{360} + \frac {x^{3}}{60} + \frac {x^{2}}{12} + \frac {x}{3} + 1\right ) + \frac {C_{1} \left (x + 1\right )}{x^{2}} + O\left (x^{6}\right ) \]