20.25.4 problem 5

Internal problem ID [4009]
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 11.4. page 758
Problem number : 5
Date solved : Sunday, March 30, 2025 at 02:14:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✗ Maple

Order:=6; 
ode:=diff(diff(y(x),x),x)+2/x/(x-3)*diff(y(x),x)-1/x^3/(x+3)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.207 (sec). Leaf size: 258

ode=D[y[x],{x,2}]+2/(x*(x-3))*D[y[x],x]-1/(x^3*(x+3))*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 e^{-\frac {2}{\sqrt {3} \sqrt {x}}} \left (\frac {10879996003390494539 x^{9/2}}{6059672463464202240 \sqrt {3}}+\frac {64713480610417 x^{7/2}}{328758271672320 \sqrt {3}}+\frac {287821451 x^{5/2}}{3397386240 \sqrt {3}}+\frac {19817 x^{3/2}}{73728 \sqrt {3}}-\frac {4894564486149401320457 x^5}{1246561192484064460800}-\frac {116612812982297797 x^4}{378729528966512640}-\frac {22160647459 x^3}{587068342272}+\frac {463507 x^2}{42467328}+\frac {587 x}{4608}+\frac {25 \sqrt {x}}{16 \sqrt {3}}+1\right ) x^{13/12}+c_2 e^{\frac {2}{\sqrt {3} \sqrt {x}}} \left (-\frac {10879996003390494539 x^{9/2}}{6059672463464202240 \sqrt {3}}-\frac {64713480610417 x^{7/2}}{328758271672320 \sqrt {3}}-\frac {287821451 x^{5/2}}{3397386240 \sqrt {3}}-\frac {19817 x^{3/2}}{73728 \sqrt {3}}-\frac {4894564486149401320457 x^5}{1246561192484064460800}-\frac {116612812982297797 x^4}{378729528966512640}-\frac {22160647459 x^3}{587068342272}+\frac {463507 x^2}{42467328}+\frac {587 x}{4608}-\frac {25 \sqrt {x}}{16 \sqrt {3}}+1\right ) x^{13/12} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + 2*Derivative(y(x), x)/(x*(x - 3)) - y(x)/(x**3*(x + 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE Derivative(y(x), (x, 2)) + 2*Derivative(y(x), x)/(x*(x - 3)) - y(x)/(x**3*(x + 3)) does not match hint 2nd_power_series_regular