46.5.7 problem 17

Internal problem ID [7343]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. REVIEW QUESTIONS. page 201
Problem number : 17
Date solved : Sunday, March 30, 2025 at 11:54:52 AM
CAS classification : [_Laguerre]

\begin{align*} x y^{\prime \prime }-\left (x +1\right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 44

Order:=6; 
ode:=x*diff(diff(y(x),x),x)-(1+x)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 66

ode=x*D[y[x],{x,2}]-(x+1)*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right )+c_2 \left (\frac {x^6}{360}+\frac {x^5}{60}+\frac {x^4}{12}+\frac {x^3}{3}+x^2\right ) \]

✓ Sympy. Time used: 0.899 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) - (x + 1)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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