12.14.5 problem 2

Internal problem ID [1946]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 2
Date solved : Tuesday, March 04, 2025 at 01:46:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} 3 x^{2} y^{\prime \prime }+2 x \left (-2 x^{2}+x +1\right ) y^{\prime }+\left (-8 x^{2}+2 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 44

Order:=6; 
ode:=3*x^2*diff(diff(y(x),x),x)+2*x*(-2*x^2+x+1)*diff(y(x),x)+(-8*x^2+2*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{1}/{3}} \left (1-\frac {2}{3} x +\frac {8}{9} x^{2}-\frac {40}{81} x^{3}+\frac {92}{243} x^{4}-\frac {664}{3645} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-x +\frac {6}{5} x^{2}-\frac {4}{5} x^{3}+\frac {32}{55} x^{4}-\frac {24}{77} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 83

ode=3*x^2*D[y[x],{x,2}]+2*x*(1+x-2*x^2)*D[y[x],x]+(2*x-8*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {664 x^5}{3645}+\frac {92 x^4}{243}-\frac {40 x^3}{81}+\frac {8 x^2}{9}-\frac {2 x}{3}+1\right )+c_2 \left (-\frac {24 x^5}{77}+\frac {32 x^4}{55}-\frac {4 x^3}{5}+\frac {6 x^2}{5}-x+1\right ) \]

✓ Sympy. Time used: 1.339 (sec). Leaf size: 71

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*Derivative(y(x), (x, 2)) + 2*x*(-2*x**2 + x + 1)*Derivative(y(x), x) + (-8*x**2 + 2*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {24 x^{5}}{77} + \frac {32 x^{4}}{55} - \frac {4 x^{3}}{5} + \frac {6 x^{2}}{5} - x + 1\right ) + C_{1} \sqrt [3]{x} \left (\frac {92 x^{4}}{243} - \frac {40 x^{3}}{81} + \frac {8 x^{2}}{9} - \frac {2 x}{3} + 1\right ) + O\left (x^{6}\right ) \]