54.9.25 problem 26

Internal problem ID [8695]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number : 26
Date solved : Sunday, March 30, 2025 at 01:23:48 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 52

Order:=8; 
ode:=x*diff(diff(y(x),x),x)+(2-x)*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1+\frac {1}{2} x +\frac {1}{6} x^{2}+\frac {1}{24} x^{3}+\frac {1}{120} x^{4}+\frac {1}{720} x^{5}+\frac {1}{5040} x^{6}+\frac {1}{40320} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_2 \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.076 (sec). Leaf size: 90

ode=x*D[y[x],{x,2}]+(2-x)*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {x^5}{720}+\frac {x^4}{120}+\frac {x^3}{24}+\frac {x^2}{6}+\frac {x}{2}+\frac {1}{x}+1\right )+c_2 \left (\frac {x^6}{5040}+\frac {x^5}{720}+\frac {x^4}{120}+\frac {x^3}{24}+\frac {x^2}{6}+\frac {x}{2}+1\right ) \]

✓ Sympy. Time used: 0.834 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (2 - x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{7}}{40320} + \frac {x^{6}}{5040} + \frac {x^{5}}{720} + \frac {x^{4}}{120} + \frac {x^{3}}{24} + \frac {x^{2}}{6} + \frac {x}{2} + 1\right ) + \frac {C_{1}}{x} + O\left (x^{8}\right ) \]