12.14.53 problem 64

Internal problem ID [1994]
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 : 64
Date solved : Tuesday, March 04, 2025 at 01:47:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} \left (5+x \right ) y^{\prime \prime }+9 x \left (5+9 x \right ) y^{\prime }-\left (5-8 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:=9*x^2*(5+x)*diff(diff(y(x),x),x)+9*x*(5+9*x)*diff(y(x),x)-(5-8*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{2}/{3}} \left (1-\frac {11}{25} x +\frac {11}{50} x^{2}-\frac {1}{10} x^{3}+\frac {29}{700} x^{4}-\frac {4727}{297500} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+x -\frac {1}{2} x^{2}+\frac {17}{70} x^{3}-\frac {187}{1750} x^{4}+\frac {24497}{568750} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{3}}} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 86

ode=9*x^2*(5+x)*D[y[x],{x,2}]+9*x*(5+9*x)*D[y[x],x]-(5-8*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {4727 x^5}{297500}+\frac {29 x^4}{700}-\frac {x^3}{10}+\frac {11 x^2}{50}-\frac {11 x}{25}+1\right )+\frac {c_2 \left (\frac {24497 x^5}{568750}-\frac {187 x^4}{1750}+\frac {17 x^3}{70}-\frac {x^2}{2}+x+1\right )}{\sqrt [3]{x}} \]

✓ Sympy. Time used: 1.137 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x**2*(x + 5)*Derivative(y(x), (x, 2)) + 9*x*(9*x + 5)*Derivative(y(x), x) - (5 - 8*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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