12.15.11 problem 7

Internal problem ID [2009]
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 : 7
Date solved : Saturday, March 29, 2025 at 11:45:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 52

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-2 x +\frac {7}{4} x^{2}-\frac {7}{9} x^{3}+\frac {77}{576} x^{4}+\frac {217}{7200} x^{5}-\frac {8813}{518400} x^{6}+\frac {143}{453600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (3 x -\frac {15}{4} x^{2}+\frac {239}{108} x^{3}-\frac {2021}{3456} x^{4}-\frac {1241}{54000} x^{5}+\frac {93859}{1728000} x^{6}-\frac {311177}{42336000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 153

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

\[ y(x)\to c_1 \left (\frac {143 x^7}{453600}-\frac {8813 x^6}{518400}+\frac {217 x^5}{7200}+\frac {77 x^4}{576}-\frac {7 x^3}{9}+\frac {7 x^2}{4}-2 x+1\right )+c_2 \left (-\frac {311177 x^7}{42336000}+\frac {93859 x^6}{1728000}-\frac {1241 x^5}{54000}-\frac {2021 x^4}{3456}+\frac {239 x^3}{108}-\frac {15 x^2}{4}+\left (\frac {143 x^7}{453600}-\frac {8813 x^6}{518400}+\frac {217 x^5}{7200}+\frac {77 x^4}{576}-\frac {7 x^3}{9}+\frac {7 x^2}{4}-2 x+1\right ) \log (x)+3 x\right ) \]

✓ Sympy. Time used: 1.153 (sec). Leaf size: 53

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

\[ y{\left (x \right )} = C_{1} \left (\frac {143 x^{7}}{453600} - \frac {8813 x^{6}}{518400} + \frac {217 x^{5}}{7200} + \frac {77 x^{4}}{576} - \frac {7 x^{3}}{9} + \frac {7 x^{2}}{4} - 2 x + 1\right ) + O\left (x^{8}\right ) \]