6.15.31 problem 27

Internal problem ID [2029]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS II. Exercises 7.6. Page 374
Problem number : 27
Date solved : Tuesday, September 30, 2025 at 05:22:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (7+x \right ) y^{\prime }+\left (9-x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 44

Order:=8; 
ode:=x^2*(1-x)*diff(diff(y(x),x),x)+x*(7+x)*diff(y(x),x)+(9-x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+16 x +36 x^{2}+16 x^{3}+x^{4}+\operatorname {O}\left (x^{8}\right )\right )+\left (\left (-40\right ) x -150 x^{2}-\frac {280}{3} x^{3}-\frac {25}{3} x^{4}+\operatorname {O}\left (x^{8}\right )\right ) c_2}{x^{3}} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 83

ode=x^2*(1-x)*D[y[x],{x,2}]+x*(7+x)*D[y[x],x]+(9-x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to \frac {c_1 \left (x^4+16 x^3+36 x^2+16 x+1\right )}{x^3}+c_2 \left (\frac {-\frac {25 x^4}{3}-\frac {280 x^3}{3}-150 x^2-40 x}{x^3}+\frac {\left (x^4+16 x^3+36 x^2+16 x+1\right ) \log (x)}{x^3}\right ) \]

✓ Sympy. Time used: 0.396 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - x)*Derivative(y(x), (x, 2)) + x*(x + 7)*Derivative(y(x), x) + (9 - x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

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