6.16.7 problem 3

Internal problem ID [2069]
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 : 3
Date solved : Tuesday, September 30, 2025 at 05:23:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 46

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

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

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 53

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

\[ y(x)\to c_1 \left (\sqrt {x} \log (x)-\frac {2 x^4-3 x^3+6 x^2+6 x-6}{6 \sqrt {x}}\right )+c_2 \sqrt {x} \]

✓ Sympy. Time used: 0.411 (sec). Leaf size: 19

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

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