20.26.28 problem 22

Internal problem ID [4053]
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 11.5. page 771
Problem number : 22
Date solved : Tuesday, March 04, 2025 at 05:23:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (5-x \right ) y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 42

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*(5-x)*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-2 x +\frac {1}{2} x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\left (5 x -\frac {9}{4} x^{2}+\frac {1}{18} x^{3}+\frac {1}{288} x^{4}+\frac {1}{3600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x^{2}} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 80

ode=x^2*D[y[x],{x,2}]+x*(5-x)*D[y[x],x]+4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_1 \left (\frac {x^2}{2}-2 x+1\right )}{x^2}+c_2 \left (\frac {\left (\frac {x^2}{2}-2 x+1\right ) \log (x)}{x^2}+\frac {\frac {x^5}{3600}+\frac {x^4}{288}+\frac {x^3}{18}-\frac {9 x^2}{4}+5 x}{x^2}\right ) \]

✓ Sympy. Time used: 0.818 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*(5 - x)*Derivative(y(x), x) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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