52.2.27 problem 27

Internal problem ID [8278]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS ABOUT SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number : 27
Date solved : Wednesday, March 05, 2025 at 05:33:55 AM
CAS classification : [_Laguerre, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x y^{\prime \prime }-x y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 46

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

\[ y = \ln \left (x \right ) \left (-x +\operatorname {O}\left (x^{8}\right )\right ) c_{2} +c_{1} x \left (1+\operatorname {O}\left (x^{8}\right )\right )+\left (1+x -\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {1}{72} x^{4}-\frac {1}{480} x^{5}-\frac {1}{3600} x^{6}-\frac {1}{30240} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

✓ Mathematica. Time used: 0.083 (sec). Leaf size: 51

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

\[ y(x)\to c_1 \left (\frac {-2 x^6-15 x^5-100 x^4-600 x^3-3600 x^2+14400 x+7200}{7200}-x \log (x)\right )+c_2 x \]

✓ Sympy. Time used: 0.751 (sec). Leaf size: 8

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

\[ y{\left (x \right )} = C_{1} x + O\left (x^{8}\right ) \]