12.14.56 problem 67

Internal problem ID [1997]
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 : 67
Date solved : Tuesday, March 04, 2025 at 01:47:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} \left (1+x \right )^{2} y^{\prime \prime }-x \left (-11 x^{2}-10 x +1\right ) y^{\prime }+\left (5 x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 44

Order:=6; 
ode:=3*x^2*(1+x)^2*diff(diff(y(x),x),x)-x*(-11*x^2-10*x+1)*diff(y(x),x)+(5*x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 66

ode=3*x^2*(1+x)^2*D[y[x],{x,2}]-x*(1-10*x-11*x^2)*D[y[x],x]+(1+5*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 1.069 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*(x + 1)**2*Derivative(y(x), (x, 2)) - x*(-11*x**2 - 10*x + 1)*Derivative(y(x), x) + (5*x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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