15.24.11 problem 11

Internal problem ID [3383]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 42, page 206
Problem number : 11
Date solved : Tuesday, March 04, 2025 at 04:37:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-9 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 43

Order:=6; 
ode:=x^2*(1-x)*diff(diff(y(x),x),x)+x*(1+x)*diff(y(x),x)-9*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = c_{1} x^{3} \left (1+\frac {3}{7} x +\frac {3}{14} x^{2}+\frac {5}{42} x^{3}+\frac {1}{14} x^{4}+\frac {1}{22} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-86400+259200 x -259200 x^{2}+86400 x^{3}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 57

ode=x^2*(1-x)*D[y[x],{x,2}]+x*(1+x)*D[y[x],x]-9*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{x^3}-\frac {3}{x^2}+\frac {3}{x}-1\right )+c_2 \left (\frac {x^7}{14}+\frac {5 x^6}{42}+\frac {3 x^5}{14}+\frac {3 x^4}{7}+x^3\right ) \]

✓ Sympy. Time used: 1.050 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - x)*Derivative(y(x), (x, 2)) + x*(x + 1)*Derivative(y(x), x) - 9*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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