7.16.4 problem 4

Internal problem ID [501]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.4 (Method of Frobenius: The exceptional cases). Problems at page 246
Problem number : 4
Date solved : Saturday, March 29, 2025 at 04:55:12 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} 5 x y^{\prime \prime }+\left (30+3 x \right ) y^{\prime }+3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 44

Order:=6; 
ode:=5*x*diff(diff(y(x),x),x)+(30+3*x)*diff(y(x),x)+3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1-\frac {1}{10} x +\frac {3}{350} x^{2}-\frac {9}{14000} x^{3}+\frac {3}{70000} x^{4}-\frac {9}{3500000} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (2880-1728 x +\frac {2592}{5} x^{2}-\frac {2592}{25} x^{3}+\frac {1944}{125} x^{4}-\frac {5832}{3125} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{5}} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 70

ode=5*x*D[y[x],{x,2}]+(30+3*x)*D[y[x],x]+3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {3 x^4}{70000}-\frac {9 x^3}{14000}+\frac {3 x^2}{350}-\frac {x}{10}+1\right )+c_1 \left (\frac {1}{x^5}-\frac {3}{5 x^4}+\frac {9}{50 x^3}-\frac {9}{250 x^2}+\frac {27}{5000 x}\right ) \]

✓ Sympy. Time used: 0.973 (sec). Leaf size: 71

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(5*x*Derivative(y(x), (x, 2)) + (3*x + 30)*Derivative(y(x), x) + 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {9 x^{5}}{3500000} + \frac {3 x^{4}}{70000} - \frac {9 x^{3}}{14000} + \frac {3 x^{2}}{350} - \frac {x}{10} + 1\right ) + \frac {C_{1} \left (\frac {27 x^{4}}{5000} - \frac {9 x^{3}}{250} + \frac {9 x^{2}}{50} - \frac {3 x}{5} + 1\right )}{x^{5}} + O\left (x^{6}\right ) \]