12.14.55 problem 66

Internal problem ID [1996]
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 : 66
Date solved : Tuesday, March 04, 2025 at 01:47:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 46

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

\[ y = \sqrt {x}\, \left (x \left (1-3 x +8 x^{2}-21 x^{3}+55 x^{4}-144 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_1 +\left (1-6 x +17 x^{2}-45 x^{3}+118 x^{4}-309 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 78

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

\[ y(x)\to c_1 \left (55 x^{9/2}-21 x^{7/2}+8 x^{5/2}-3 x^{3/2}+\sqrt {x}\right )+c_2 \left (55 x^{11/2}-21 x^{9/2}+8 x^{7/2}-3 x^{5/2}+x^{3/2}\right ) \]

✓ Sympy. Time used: 1.169 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = C_{2} x^{\frac {3}{2}} + C_{1} \sqrt {x} + O\left (x^{6}\right ) \]