54.9.2 problem 2

Internal problem ID [8672]
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 : 2
Date solved : Sunday, March 30, 2025 at 01:23:08 PM
CAS classification : [_Laguerre]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 70

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

\[ y = c_1 \,x^{3} \left (1+\frac {5}{4} x +\frac {3}{4} x^{2}+\frac {7}{24} x^{3}+\frac {1}{12} x^{4}+\frac {3}{160} x^{5}+\frac {1}{288} x^{6}+\frac {11}{20160} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (\ln \left (x \right ) \left (24 x^{3}+30 x^{4}+18 x^{5}+7 x^{6}+2 x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (12-12 x +18 x^{2}+26 x^{3}+x^{4}-9 x^{5}-6 x^{6}-\frac {9}{4} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.094 (sec). Leaf size: 115

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

\[ y(x)\to c_1 \left (\frac {1}{12} \left (7 x^3+18 x^2+30 x+24\right ) x^3 \log (x)+\frac {1}{36} \left (-25 x^6-45 x^5-27 x^4+54 x^3+54 x^2-36 x+36\right )\right )+c_2 \left (\frac {x^9}{288}+\frac {3 x^8}{160}+\frac {x^7}{12}+\frac {7 x^6}{24}+\frac {3 x^5}{4}+\frac {5 x^4}{4}+x^3\right ) \]

✓ Sympy. Time used: 0.777 (sec). Leaf size: 36

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

\[ y{\left (x \right )} = C_{1} x^{3} \left (\frac {x^{4}}{12} + \frac {7 x^{3}}{24} + \frac {3 x^{2}}{4} + \frac {5 x}{4} + 1\right ) + O\left (x^{8}\right ) \]