6.16.17 problem 13

Internal problem ID [2079]
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 : 13
Date solved : Tuesday, September 30, 2025 at 05:23:34 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 40

Order:=6; 
ode:=x*(1+x)*diff(diff(y(x),x),x)-4*diff(y(x),x)-2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{5} \left (1-3 x +6 x^{2}-10 x^{3}+15 x^{4}-21 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (2880-1440 x +480 x^{2}-480 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 48

ode=x*(1+x)*D[y[x],{x,2}]-4*D[y[x],x]-2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^2}{6}-\frac {x}{2}+1\right )+c_2 \left (15 x^9-10 x^8+6 x^7-3 x^6+x^5\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x + 1)*Derivative(y(x), (x, 2)) - 2*y(x) - 4*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : Expected Expr or iterable but got None