20.26.25 problem 19

Internal problem ID [4050]
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 : 19
Date solved : Tuesday, March 04, 2025 at 05:23:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x \left (x +3\right ) y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 48

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-x*(3+x)*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = x^{2} \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+2 x +\frac {3}{2} x^{2}+\frac {2}{3} x^{3}+\frac {5}{24} x^{4}+\frac {1}{20} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-3\right ) x -\frac {13}{4} x^{2}-\frac {31}{18} x^{3}-\frac {173}{288} x^{4}-\frac {187}{1200} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 122

ode=x^2*D[y[x],{x,2}]-x*(x+3)*D[y[x],x]+4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^5}{20}+\frac {5 x^4}{24}+\frac {2 x^3}{3}+\frac {3 x^2}{2}+2 x+1\right ) x^2+c_2 \left (\left (-\frac {187 x^5}{1200}-\frac {173 x^4}{288}-\frac {31 x^3}{18}-\frac {13 x^2}{4}-3 x\right ) x^2+\left (\frac {x^5}{20}+\frac {5 x^4}{24}+\frac {2 x^3}{3}+\frac {3 x^2}{2}+2 x+1\right ) x^2 \log (x)\right ) \]

✓ Sympy. Time used: 0.977 (sec). Leaf size: 29

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

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