6.12.34 problem 41

Internal problem ID [1888]
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 : 41
Date solved : Tuesday, September 30, 2025 at 05:20:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+x^{6} y^{\prime }+7 x^{5} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 11

Order:=6; 
ode:=diff(diff(y(x),x),x)+x^6*diff(y(x),x)+7*x^5*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=D[y[x],{x,2}]+x^6*D[y[x],x]+7*x^5*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.227 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**6*Derivative(y(x), x) + 7*x**5*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

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