12.14.2 problem Example 7.5.2 page 354

Internal problem ID [1943]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : Example 7.5.2 page 354
Date solved : Tuesday, March 04, 2025 at 01:46:37 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x^{2} \left (3+x \right ) y^{\prime \prime }+5 x \left (1+x \right ) y^{\prime }-\left (1-4 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 48

Order:=6; 
ode:=x^2*(x+3)*diff(diff(y(x),x),x)+5*x*(1+x)*diff(y(x),x)-(1-4*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{4}/{3}} \left (1-\frac {7}{9} x +\frac {35}{81} x^{2}-\frac {455}{2187} x^{3}+\frac {1820}{19683} x^{4}-\frac {6916}{177147} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+x -x^{2}+\frac {3}{5} x^{3}-\frac {3}{10} x^{4}+\frac {3}{22} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 82

ode=x^2*(3+x)*D[y[x],{x,2}]+5*x*(1+x)*D[y[x],x]-(1-4*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {6916 x^5}{177147}+\frac {1820 x^4}{19683}-\frac {455 x^3}{2187}+\frac {35 x^2}{81}-\frac {7 x}{9}+1\right )+\frac {c_2 \left (\frac {3 x^5}{22}-\frac {3 x^4}{10}+\frac {3 x^3}{5}-x^2+x+1\right )}{x} \]

✓ Sympy. Time used: 1.068 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x + 3)*Derivative(y(x), (x, 2)) + 5*x*(x + 1)*Derivative(y(x), x) - (1 - 4*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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