12.16.18 problem 14

Internal problem ID [2080]
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 III. Exercises 7.7. Page 389
Problem number : 14
Date solved : Saturday, March 29, 2025 at 11:47:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+2 x \right ) y^{\prime \prime }+x \left (9+13 x \right ) y^{\prime }+\left (7+5 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 41

Order:=6; 
ode:=x^2*(1+2*x)*diff(diff(y(x),x),x)+x*(9+13*x)*diff(y(x),x)+(7+5*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1+\frac {4}{7} x -\frac {5}{28} x^{2}+\frac {5}{42} x^{3}-\frac {5}{48} x^{4}+\frac {7}{66} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x}+\frac {c_2 \left (-86400-449280 x -617760 x^{2}+\operatorname {O}\left (x^{6}\right )\right )}{x^{7}} \]

✓ Mathematica. Time used: 0.055 (sec). Leaf size: 54

ode=x^2*(1+2*x)*D[y[x],{x,2}]+x*(9+13*x)*D[y[x],x]+(7+5*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (-\frac {5 x^3}{48}+\frac {5 x^2}{42}-\frac {5 x}{28}+\frac {1}{x}+\frac {4}{7}\right )+c_1 \left (\frac {1}{x^7}+\frac {26}{5 x^6}+\frac {143}{20 x^5}\right ) \]

✓ Sympy. Time used: 1.034 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(2*x + 1)*Derivative(y(x), (x, 2)) + x*(13*x + 9)*Derivative(y(x), x) + (5*x + 7)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {C_{2}}{x} + \frac {C_{1}}{x^{7}} + O\left (x^{6}\right ) \]