6.15.9 problem 5

Internal problem ID [2007]
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 : 5
Date solved : Tuesday, September 30, 2025 at 05:22:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }-x \left (-2 x^{2}-4 x +1\right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 54

Order:=8; 
ode:=x^2*(x^2+x+1)*diff(diff(y(x),x),x)-x*(-2*x^2-4*x+1)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = x \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-4 x +\frac {19}{2} x^{2}-\frac {49}{3} x^{3}+\frac {515}{24} x^{4}-\frac {319}{15} x^{5}+\frac {10093}{720} x^{6}-\frac {647}{360} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (3 x -\frac {43}{4} x^{2}+\frac {208}{9} x^{3}-\frac {10379}{288} x^{4}+\frac {76321}{1800} x^{5}-\frac {172499}{4800} x^{6}+\frac {39091}{2400} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 158

ode=x^2*(1+x+x^2)*D[y[x],{x,2}]-x*(1-4*x-2*x^2)*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 x \left (-\frac {647 x^7}{360}+\frac {10093 x^6}{720}-\frac {319 x^5}{15}+\frac {515 x^4}{24}-\frac {49 x^3}{3}+\frac {19 x^2}{2}-4 x+1\right )+c_2 \left (x \left (\frac {39091 x^7}{2400}-\frac {172499 x^6}{4800}+\frac {76321 x^5}{1800}-\frac {10379 x^4}{288}+\frac {208 x^3}{9}-\frac {43 x^2}{4}+3 x\right )+x \left (-\frac {647 x^7}{360}+\frac {10093 x^6}{720}-\frac {319 x^5}{15}+\frac {515 x^4}{24}-\frac {49 x^3}{3}+\frac {19 x^2}{2}-4 x+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 1.617 (sec). Leaf size: 8

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 + x + 1)*Derivative(y(x), (x, 2)) - x*(-2*x**2 - 4*x + 1)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

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