15.23.6 problem 6

Internal problem ID [3356]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 41, page 195
Problem number : 6
Date solved : Tuesday, March 04, 2025 at 04:36:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} y^{\prime \prime }+\left (2+3 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 47

Order:=6; 
ode:=9*x^2*diff(diff(y(x),x),x)+(2+3*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = c_{1} x^{{1}/{3}} \left (1-\frac {1}{2} x +\frac {1}{20} x^{2}-\frac {1}{480} x^{3}+\frac {1}{21120} x^{4}-\frac {1}{1478400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{{2}/{3}} \left (1-\frac {1}{4} x +\frac {1}{56} x^{2}-\frac {1}{1680} x^{3}+\frac {1}{87360} x^{4}-\frac {1}{6988800} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 90

ode=9*x^2*D[y[x],{x,2}]+(2+3*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \sqrt [3]{x} \left (-\frac {x^5}{1478400}+\frac {x^4}{21120}-\frac {x^3}{480}+\frac {x^2}{20}-\frac {x}{2}+1\right )+c_1 x^{2/3} \left (-\frac {x^5}{6988800}+\frac {x^4}{87360}-\frac {x^3}{1680}+\frac {x^2}{56}-\frac {x}{4}+1\right ) \]

✓ Sympy. Time used: 0.845 (sec). Leaf size: 60

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x**2*Derivative(y(x), (x, 2)) + (3*x + 2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {2}{3}} \left (\frac {x^{4}}{87360} - \frac {x^{3}}{1680} + \frac {x^{2}}{56} - \frac {x}{4} + 1\right ) + C_{1} \sqrt [3]{x} \left (\frac {x^{4}}{21120} - \frac {x^{3}}{480} + \frac {x^{2}}{20} - \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]