6.15.8 problem 4

Internal problem ID [2006]
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 : 4
Date solved : Tuesday, September 30, 2025 at 05:22:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+12 x^{2} \left (1+x \right ) y^{\prime }+\left (3 x^{2}+3 x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 56

Order:=8; 
ode:=4*x^2*(x^2+x+1)*diff(diff(y(x),x),x)+12*x^2*(1+x)*diff(y(x),x)+(3*x^2+3*x+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-2 x +\frac {5}{2} x^{2}-2 x^{3}+\frac {5}{8} x^{4}+\frac {17}{20} x^{5}-\frac {121}{80} x^{6}+x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (x -\frac {9}{4} x^{2}+\frac {17}{6} x^{3}-\frac {205}{96} x^{4}+\frac {481}{1200} x^{5}+\frac {2109}{1600} x^{6}-\frac {1063}{560} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \right ) \sqrt {x} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 156

ode=4*x^2*(1+x+x^2)*D[y[x],{x,2}]+12*x^2*(1+x)*D[y[x],x]+(1+3*x+3*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \sqrt {x} \left (x^7-\frac {121 x^6}{80}+\frac {17 x^5}{20}+\frac {5 x^4}{8}-2 x^3+\frac {5 x^2}{2}-2 x+1\right )+c_2 \left (\sqrt {x} \left (-\frac {1063 x^7}{560}+\frac {2109 x^6}{1600}+\frac {481 x^5}{1200}-\frac {205 x^4}{96}+\frac {17 x^3}{6}-\frac {9 x^2}{4}+x\right )+\sqrt {x} \left (x^7-\frac {121 x^6}{80}+\frac {17 x^5}{20}+\frac {5 x^4}{8}-2 x^3+\frac {5 x^2}{2}-2 x+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 1.630 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(12*x**2*(x + 1)*Derivative(y(x), x) + 4*x**2*(x**2 + x + 1)*Derivative(y(x), (x, 2)) + (3*x**2 + 3*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} \sqrt {x} + O\left (x^{8}\right ) \]