12.12.36 problem 43

Internal problem ID [1890]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 43
Date solved : Tuesday, March 04, 2025 at 01:45:43 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (-x^{6}+1\right ) y^{\prime \prime }-12 x^{5} y^{\prime }-30 x^{4} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 15

Order:=6; 
ode:=(-x^6+1)*diff(diff(y(x),x),x)-12*x^5*diff(y(x),x)-30*x^4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 10

ode=(1-x^6)*D[y[x],{x,2}]-12*x^5*D[y[x],x]-30*x^4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 x+c_1 \]

✓ Sympy. Time used: 0.722 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-12*x**5*Derivative(y(x), x) - 30*x**4*y(x) + (1 - x**6)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

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