6.15.57 problem 58

Internal problem ID [2055]
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 : 58
Date solved : Tuesday, September 30, 2025 at 05:23:05 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (1+x \right ) y^{\prime \prime }+8 x^{2} y^{\prime }+\left (1+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 40

Order:=6; 
ode:=4*x^2*(1+x)*diff(diff(y(x),x),x)+8*x^2*diff(y(x),x)+(1+x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 64

ode=4*x^2*(1+x)*D[y[x],{x,2}]+8*x^2*D[y[x],x]+(1+x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.294 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*(x + 1)*Derivative(y(x), (x, 2)) + 8*x**2*Derivative(y(x), x) + (x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \sqrt {x} \left (\frac {35 x^{4}}{192} - \frac {5 x^{3}}{12} + \frac {3 x^{2}}{4} - x + 1\right ) + O\left (x^{6}\right ) \]