12.15.58 problem 59

Internal problem ID [2056]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS II. Exercises 7.6. Page 374
Problem number : 59
Date solved : Tuesday, March 04, 2025 at 01:49:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} \left (3+x \right ) y^{\prime \prime }+3 x \left (3+7 x \right ) y^{\prime }+\left (3+4 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 44

Order:=6; 
ode:=9*x^2*(x+3)*diff(diff(y(x),x),x)+3*x*(3+7*x)*diff(y(x),x)+(3+4*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x^{{1}/{3}} \left (1-\frac {1}{3} x +\frac {1}{9} x^{2}-\frac {1}{27} x^{3}+\frac {1}{81} x^{4}-\frac {1}{243} x^{5}\right ) \left (c_2 \ln \left (x \right )+c_1 \right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 92

ode=9*x^2*(3+x)*D[y[x],{x,2}]+3*x*(3+7*x)*D[y[x],x]+(3+4*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {x^5}{243}+\frac {x^4}{81}-\frac {x^3}{27}+\frac {x^2}{9}-\frac {x}{3}+1\right )+c_2 \sqrt [3]{x} \left (-\frac {x^5}{243}+\frac {x^4}{81}-\frac {x^3}{27}+\frac {x^2}{9}-\frac {x}{3}+1\right ) \log (x) \]

✓ Sympy. Time used: 0.928 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x**2*(x + 3)*Derivative(y(x), (x, 2)) + 3*x*(7*x + 3)*Derivative(y(x), x) + (4*x + 3)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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