12.14.26 problem 28

Internal problem ID [1967]
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 : 28
Date solved : Tuesday, March 04, 2025 at 01:47:08 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (8+x \right ) y^{\prime \prime }+x \left (2+3 x \right ) y^{\prime }+\left (1+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 47

Order:=6; 
ode:=x^2*(8+x)*diff(diff(y(x),x),x)+x*(3*x+2)*diff(y(x),x)+(1+x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{1}/{4}} \left (1-\frac {25}{96} x +\frac {675}{14336} x^{2}-\frac {38025}{5046272} x^{3}+\frac {732615}{645922816} x^{4}-\frac {9230949}{56103010304} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \sqrt {x}\, \left (1-\frac {9}{40} x +\frac {5}{128} x^{2}-\frac {245}{39936} x^{3}+\frac {6615}{7241728} x^{4}-\frac {7623}{57933824} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 90

ode=x^2*(8+x)*D[y[x],{x,2}]+x*(2+3*x)*D[y[x],x]+(1+x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {7623 x^5}{57933824}+\frac {6615 x^4}{7241728}-\frac {245 x^3}{39936}+\frac {5 x^2}{128}-\frac {9 x}{40}+1\right )+c_2 \sqrt [4]{x} \left (-\frac {9230949 x^5}{56103010304}+\frac {732615 x^4}{645922816}-\frac {38025 x^3}{5046272}+\frac {675 x^2}{14336}-\frac {25 x}{96}+1\right ) \]

✓ Sympy. Time used: 0.948 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x + 8)*Derivative(y(x), (x, 2)) + x*(3*x + 2)*Derivative(y(x), x) + (x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt {x} + C_{1} \sqrt [4]{x} + O\left (x^{6}\right ) \]