6.15.49 problem 45

Internal problem ID [2047]
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 : 45
Date solved : Tuesday, September 30, 2025 at 05:22:59 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 28

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x +\operatorname {O}\left (x^{6}\right )\right )+\left (4 x +\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 27

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

\[ y(x)\to c_1 (1-x)+c_2 (4 x+(1-x) \log (x)) \]

✓ Sympy. Time used: 0.321 (sec). Leaf size: 7

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x + 1)*Derivative(y(x), (x, 2)) + (1 - 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)
 

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