20.24.16 problem Problem 17

Internal problem ID [4001]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.2. page 739
Problem number : Problem 17
Date solved : Sunday, March 30, 2025 at 02:14:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 40

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {1}{4} x^{2}+\frac {1}{18} x^{3}+\frac {5}{192} x^{4}+\frac {23}{3600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (x +\frac {11}{108} x^{3}+\frac {11}{1152} x^{4}+\frac {883}{216000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 96

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

\[ y(x)\to c_1 \left (\frac {23 x^5}{3600}+\frac {5 x^4}{192}+\frac {x^3}{18}+\frac {x^2}{4}+1\right )+c_2 \left (\frac {883 x^5}{216000}+\frac {11 x^4}{1152}+\frac {11 x^3}{108}+\left (\frac {23 x^5}{3600}+\frac {5 x^4}{192}+\frac {x^3}{18}+\frac {x^2}{4}+1\right ) \log (x)+x\right ) \]

✓ Sympy. Time used: 0.932 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = C_{1} \left (\frac {23 x^{5}}{3600} + \frac {5 x^{4}}{192} + \frac {x^{3}}{18} + \frac {x^{2}}{4} + 1\right ) + O\left (x^{6}\right ) \]