6.15.50 problem 46

Internal problem ID [2048]
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 : 46
Date solved : Tuesday, September 30, 2025 at 05:22:59 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 42

Order:=6; 
ode:=x^2*(1-x)*diff(diff(y(x),x),x)+x*(3-2*x)*diff(y(x),x)+(2*x+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (3 x -3 x^{2}+\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1-2 x +x^{2}+\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right )}{x} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 70

ode=x^2*(1-x)*D[y[x],{x,2}]+x*(3-2*x)*D[y[x],x]+(1+2*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_1 \left (x^2-2 x+1\right )}{x}+c_2 \left (\frac {\left (x^2-2 x+1\right ) \log (x)}{x}+\frac {\frac {x^5}{30}+\frac {x^4}{12}+\frac {x^3}{3}-3 x^2+3 x}{x}\right ) \]

✓ Sympy. Time used: 0.411 (sec). Leaf size: 8

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - x)*Derivative(y(x), (x, 2)) + x*(3 - 2*x)*Derivative(y(x), x) + (2*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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