7.15.20 problem 20

Internal problem ID [476]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 20
Date solved : Tuesday, March 04, 2025 at 11:24:58 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 3 x y^{\prime \prime }+2 y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 44

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

\[ y = c_1 \,x^{{1}/{3}} \left (1-\frac {1}{2} x +\frac {1}{14} x^{2}-\frac {1}{210} x^{3}+\frac {1}{5460} x^{4}-\frac {1}{218400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-x +\frac {1}{5} x^{2}-\frac {1}{60} x^{3}+\frac {1}{1320} x^{4}-\frac {1}{46200} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 83

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

\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {x^5}{218400}+\frac {x^4}{5460}-\frac {x^3}{210}+\frac {x^2}{14}-\frac {x}{2}+1\right )+c_2 \left (-\frac {x^5}{46200}+\frac {x^4}{1320}-\frac {x^3}{60}+\frac {x^2}{5}-x+1\right ) \]

✓ Sympy. Time used: 0.859 (sec). Leaf size: 58

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{5}}{46200} + \frac {x^{4}}{1320} - \frac {x^{3}}{60} + \frac {x^{2}}{5} - x + 1\right ) + C_{1} \sqrt [3]{x} \left (\frac {x^{4}}{5460} - \frac {x^{3}}{210} + \frac {x^{2}}{14} - \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]