6.15.7 problem 3

Internal problem ID [2005]
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 : 3
Date solved : Tuesday, September 30, 2025 at 05:22:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x^{2} \left (x^{2}+2 x +1\right ) y^{\prime \prime }+x \left (4 x^{2}+3 x +1\right ) y^{\prime }-x \left (1-2 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 52

Order:=8; 
ode:=x^2*(x^2+2*x+1)*diff(diff(y(x),x),x)+x*(4*x^2+3*x+1)*diff(y(x),x)-x*(1-2*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+x -x^{2}+\frac {1}{3} x^{3}+\frac {1}{3} x^{4}-\frac {11}{15} x^{5}+\frac {37}{45} x^{6}-\frac {209}{315} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\left (-3\right ) x +\frac {1}{2} x^{2}+\frac {31}{18} x^{3}-\frac {91}{36} x^{4}+\frac {1897}{900} x^{5}-\frac {301}{300} x^{6}-\frac {3901}{14700} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 145

ode=x^2*(1+2*x+x^2)*D[y[x],{x,2}]+x*(1+3*x+4*x^2)*D[y[x],x]-x*(1-2*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {209 x^7}{315}+\frac {37 x^6}{45}-\frac {11 x^5}{15}+\frac {x^4}{3}+\frac {x^3}{3}-x^2+x+1\right )+c_2 \left (-\frac {3901 x^7}{14700}-\frac {301 x^6}{300}+\frac {1897 x^5}{900}-\frac {91 x^4}{36}+\frac {31 x^3}{18}+\frac {x^2}{2}+\left (-\frac {209 x^7}{315}+\frac {37 x^6}{45}-\frac {11 x^5}{15}+\frac {x^4}{3}+\frac {x^3}{3}-x^2+x+1\right ) \log (x)-3 x\right ) \]

✓ Sympy. Time used: 0.576 (sec). Leaf size: 7

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 + 2*x + 1)*Derivative(y(x), (x, 2)) - x*(1 - 2*x)*y(x) + x*(4*x**2 + 3*x + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

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