12.15.48 problem 44

Internal problem ID [2046]
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 : 44
Date solved : Tuesday, March 04, 2025 at 01:48:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 34

Order:=6; 
ode:=x^2*(1-2*x)*diff(diff(y(x),x),x)+3*x*diff(y(x),x)+(1+4*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-6\right ) x +6 x^{2}-\frac {8}{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 40

ode=x^2*(1-2*x)*D[y[x],{x,2}]+3*x*D[y[x],x]+(1+4*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {-\frac {8 x^3}{3}+6 x^2-6 x}{x}+\frac {\log (x)}{x}\right )+\frac {c_1}{x} \]

✓ Sympy. Time used: 1.028 (sec). Leaf size: 8

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - 2*x)*Derivative(y(x), (x, 2)) + 3*x*Derivative(y(x), x) + (4*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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