54.6.4 problem 4

Internal problem ID [8636]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.8 Indicial Equation with Difference of Roots a Positive Integer: Nonlogarithmic Case. Exercises page 380
Problem number : 4
Date solved : Wednesday, March 05, 2025 at 06:11:09 AM
CAS classification : [_Laguerre]

\begin{align*} x y^{\prime \prime }-\left (3+x \right ) y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 50

Order:=8; 
ode:=x*diff(diff(y(x),x),x)-(x+3)*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_{1} x^{4} \left (1+\frac {2}{5} x +\frac {1}{10} x^{2}+\frac {2}{105} x^{3}+\frac {1}{336} x^{4}+\frac {1}{2520} x^{5}+\frac {1}{21600} x^{6}+\frac {1}{207900} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (-144-96 x -24 x^{2}+2 x^{4}+\frac {4}{5} x^{5}+\frac {1}{5} x^{6}+\frac {4}{105} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.102 (sec). Leaf size: 91

ode=x*D[y[x],{x,2}]-(3+x)*D[y[x],x]+2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {x^6}{720}-\frac {x^5}{180}-\frac {x^4}{72}+\frac {x^2}{6}+\frac {2 x}{3}+1\right )+c_2 \left (\frac {x^{10}}{21600}+\frac {x^9}{2520}+\frac {x^8}{336}+\frac {2 x^7}{105}+\frac {x^6}{10}+\frac {2 x^5}{5}+x^4\right ) \]

✓ Sympy. Time used: 0.887 (sec). Leaf size: 42

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{2}}{6} + \frac {2 x}{3} + 1\right ) + C_{1} x^{4} \left (\frac {2 x^{3}}{105} + \frac {x^{2}}{10} + \frac {2 x}{5} + 1\right ) + O\left (x^{8}\right ) \]