67.9.13 problem 14.2 (c)

Internal problem ID [16561]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 14. Higher order equations and the reduction of order method. Additional exercises page 277
Problem number : 14.2 (c)
Date solved : Thursday, October 02, 2025 at 01:36:28 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} y^{\prime \prime }-6 x y^{\prime }+12 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x^{3} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 13

ode:=x^2*diff(diff(y(x),x),x)-6*x*diff(y(x),x)+12*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{3} \left (c_1 x +c_2 \right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 16

ode=x^2*D[y[x],{x,2}]-6*x*D[y[x],x]+12*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x^3 (c_2 x+c_1) \end{align*}

✓ Sympy. Time used: 0.092 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - 6*x*Derivative(y(x), x) + 12*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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