6.16.8 problem 4

Internal problem ID [2070]
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 III. Exercises 7.7. Page 389
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 05:23:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 60

Order:=6; 
ode:=x*(1+x)*diff(diff(y(x),x),x)+x*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 x \left (1-x +\frac {5}{6} x^{2}-\frac {25}{36} x^{3}+\frac {85}{144} x^{4}-\frac {221}{432} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-x +x^{2}-\frac {5}{6} x^{3}+\frac {25}{36} x^{4}-\frac {85}{144} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_2 +\left (1-x +\frac {1}{2} x^{2}-\frac {7}{18} x^{3}+\frac {145}{432} x^{4}-\frac {257}{864} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 85

ode=x*(1+x)*D[y[x],{x,2}]+x*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{36} x \left (25 x^3-30 x^2+36 x-36\right ) \log (x)+\frac {1}{432} \left (-455 x^4+552 x^3-648 x^2+432 x+432\right )\right )+c_2 \left (\frac {85 x^5}{144}-\frac {25 x^4}{36}+\frac {5 x^3}{6}-x^2+x\right ) \]

✓ Sympy. Time used: 0.272 (sec). Leaf size: 10

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

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