6.14.6 problem 3

Internal problem ID [1947]
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 : 3
Date solved : Tuesday, September 30, 2025 at 05:21:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+3 x +3\right ) y^{\prime \prime }+x \left (7 x^{2}+8 x +5\right ) y^{\prime }-\left (-9 x^{2}-2 x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 44

Order:=6; 
ode:=x^2*(x^2+3*x+3)*diff(diff(y(x),x),x)+x*(7*x^2+8*x+5)*diff(y(x),x)-(-9*x^2-2*x+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{4}/{3}} \left (1-\frac {4}{7} x -\frac {7}{45} x^{2}+\frac {970}{2457} x^{3}-\frac {5707}{22680} x^{4}+\frac {13568}{300105} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1-x^{2}+\frac {2}{3} x^{3}-\frac {10}{33} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 74

ode=x^2*(3+3*x+x^2)*D[y[x],{x,2}]+x*(5+8*x+7*x^2)*D[y[x],x]-(1-2*x-9*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_2 \left (-\frac {10 x^5}{33}+\frac {2 x^3}{3}-x^2+1\right )}{x}+c_1 \sqrt [3]{x} \left (\frac {13568 x^5}{300105}-\frac {5707 x^4}{22680}+\frac {970 x^3}{2457}-\frac {7 x^2}{45}-\frac {4 x}{7}+1\right ) \]

✓ Sympy. Time used: 0.646 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 + 3*x + 3)*Derivative(y(x), (x, 2)) + x*(7*x**2 + 8*x + 5)*Derivative(y(x), x) - (-9*x**2 - 2*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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