6.12.29 problem 35

Internal problem ID [1883]
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 : 35
Date solved : Tuesday, September 30, 2025 at 05:20:53 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{3}+1\right ) y^{\prime \prime }+7 x^{2} y^{\prime }+9 x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 29

Order:=6; 
ode:=(x^3+1)*diff(diff(y(x),x),x)+7*x^2*diff(y(x),x)+9*x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {3 x^{3}}{2}\right ) y \left (0\right )+\left (x -\frac {4}{3} x^{4}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 28

ode=(1+x^3)*D[y[x],{x,2}]+7*x^2*D[y[x],x]+9*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.302 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(7*x**2*Derivative(y(x), x) + 9*x*y(x) + (x**3 + 1)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (1 - \frac {3 x^{3}}{2}\right ) + C_{1} x \left (1 - \frac {4 x^{3}}{3}\right ) + O\left (x^{6}\right ) \]