6.16.22 problem 18

Internal problem ID [2084]
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 : 18
Date solved : Tuesday, September 30, 2025 at 05:23:39 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+x \right ) y^{\prime \prime }+3 x^{2} y^{\prime }-\left (6-x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 39

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

\[ y = c_1 \,x^{3} \left (1-\frac {8}{3} x +\frac {100}{21} x^{2}-\frac {50}{7} x^{3}+\frac {175}{18} x^{4}-\frac {112}{9} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (2880+720 x +\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 53

ode=x^2*(1+x)*D[y[x],{x,2}]+3*x^2*D[y[x],x]-(6-x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{x^2}+\frac {1}{4 x}\right )+c_2 \left (\frac {175 x^7}{18}-\frac {50 x^6}{7}+\frac {100 x^5}{21}-\frac {8 x^4}{3}+x^3\right ) \]

✓ Sympy. Time used: 0.401 (sec). Leaf size: 42

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

\[ y{\left (x \right )} = C_{2} x^{3} \left (\frac {9 x^{2}}{7} - \frac {3 x}{2} + 1\right ) + \frac {C_{1} \left (\frac {3 x^{2}}{4} - \frac {3 x}{2} + 1\right )}{x^{2}} + O\left (x^{6}\right ) \]