6.12.35 problem 42

Internal problem ID [1889]
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 : 42
Date solved : Tuesday, September 30, 2025 at 05:20:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{8}+1\right ) y^{\prime \prime }-16 x^{7} y^{\prime }+72 x^{6} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 11

Order:=6; 
ode:=(x^8+1)*diff(diff(y(x),x),x)-16*x^7*diff(y(x),x)+72*x^6*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

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

ode=(1+x^8)*D[y[x],{x,2}]-16*x^7*D[y[x],x]+72*x^6*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.293 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-16*x**7*Derivative(y(x), x) + 72*x**6*y(x) + (x**8 + 1)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (1 - \frac {9 x^{8}}{7}\right ) + C_{1} x \left (1 - \frac {7 x^{8}}{9}\right ) + O\left (x^{6}\right ) \]