54.9.1 problem 1

Internal problem ID [8671]
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 : 1
Date solved : Wednesday, March 05, 2025 at 06:12:05 AM
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: 0.018 (sec). Leaf size: 68

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

\[ y = c_{1} x^{3} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (6 x^{3}+6 x^{4}+3 x^{5}+x^{6}+\frac {1}{4} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (12-6 x +6 x^{2}+11 x^{3}+5 x^{4}+x^{5}-\frac {1}{16} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.089 (sec). Leaf size: 104

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}]
 

\[ y(x)\to c_1 \left (\frac {1}{12} \left (x^3+3 x^2+6 x+6\right ) x^3 \log (x)+\frac {1}{36} \left (-x^6+9 x^4+27 x^3+18 x^2-18 x+36\right )\right )+c_2 \left (\frac {x^9}{720}+\frac {x^8}{120}+\frac {x^7}{24}+\frac {x^6}{6}+\frac {x^5}{2}+x^4+x^3\right ) \]

✓ Sympy. Time used: 0.742 (sec). Leaf size: 29

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

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