54.6.16 problem 16

Internal problem ID [8648]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.8 Indicial Equation with Difference of Roots a Positive Integer: Nonlogarithmic Case. Exercises page 380
Problem number : 16
Date solved : Wednesday, March 05, 2025 at 06:11:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 51

Order:=8; 
ode:=x^2*(4*x-1)*diff(diff(y(x),x),x)+x*(1+5*x)*diff(y(x),x)+3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_{1} x^{3} \left (1+\frac {39}{5} x +\frac {221}{5} x^{2}+221 x^{3}+\frac {16575}{16} x^{4}+\frac {224315}{48} x^{5}+\frac {493493}{24} x^{6}+\frac {711399}{8} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (-144+144 x +270 x^{4}+2106 x^{5}+11934 x^{6}+59670 x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.162 (sec). Leaf size: 80

ode=x^2*(4*x-1)*D[y[x],{x,2}]+x*(5*x+1)*D[y[x],x]+3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {663 x^5}{8}-\frac {117 x^4}{8}-\frac {15 x^3}{8}+\frac {1}{x}-1\right )+c_2 \left (\frac {493493 x^9}{24}+\frac {224315 x^8}{48}+\frac {16575 x^7}{16}+221 x^6+\frac {221 x^5}{5}+\frac {39 x^4}{5}+x^3\right ) \]

✓ Sympy. Time used: 1.068 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(4*x - 1)*Derivative(y(x), (x, 2)) + x*(5*x + 1)*Derivative(y(x), x) + 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} x^{3} + \frac {C_{1}}{x} + O\left (x^{8}\right ) \]