12.15.6 problem 2

Internal problem ID [2004]
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 : 2
Date solved : Tuesday, March 04, 2025 at 01:47:55 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x^{2} \left (2 x^{2}+x +1\right ) y^{\prime \prime }+x \left (7 x^{2}+6 x +3\right ) y^{\prime }+\left (-3 x^{2}+6 x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 56

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

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-2 x +\frac {9}{2} x^{2}-\frac {20}{3} x^{3}+\frac {173}{24} x^{4}-\frac {93}{20} x^{5}-\frac {419}{720} x^{6}+\frac {6697}{1260} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (x -\frac {15}{4} x^{2}+\frac {133}{18} x^{3}-\frac {3077}{288} x^{4}+\frac {4217}{400} x^{5}-\frac {70949}{14400} x^{6}-\frac {125221}{29400} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2}{x} \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 162

ode=x^2*(1+x+2*x^2)*D[y[x],{x,2}]+x*(3+6*x+7*x^2)*D[y[x],x]+(1+6*x-3*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to \frac {c_1 \left (\frac {6697 x^7}{1260}-\frac {419 x^6}{720}-\frac {93 x^5}{20}+\frac {173 x^4}{24}-\frac {20 x^3}{3}+\frac {9 x^2}{2}-2 x+1\right )}{x}+c_2 \left (\frac {-\frac {125221 x^7}{29400}-\frac {70949 x^6}{14400}+\frac {4217 x^5}{400}-\frac {3077 x^4}{288}+\frac {133 x^3}{18}-\frac {15 x^2}{4}+x}{x}+\frac {\left (\frac {6697 x^7}{1260}-\frac {419 x^6}{720}-\frac {93 x^5}{20}+\frac {173 x^4}{24}-\frac {20 x^3}{3}+\frac {9 x^2}{2}-2 x+1\right ) \log (x)}{x}\right ) \]

✓ Sympy. Time used: 1.278 (sec). Leaf size: 8

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

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