87.14.3 problem 3

Internal problem ID [23522]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 109
Problem number : 3
Date solved : Thursday, October 02, 2025 at 09:42:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime }+\left (5 x^{3}-x^{2}\right ) y^{\prime }+2 \left (3 x^{3}-x^{2}\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{-2 x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 20

ode:=x^3*diff(diff(y(x),x),x)+(5*x^3-x^2)*diff(y(x),x)+2*(3*x^3-x^2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,{\mathrm e}^{-2 x}+c_2 \,{\mathrm e}^{-3 x} \left (x +1\right ) \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 25

ode=x^3*D[y[x],{x,2}]+(5*x^3-x^2)*D[y[x],x]+2*(3*x^3-x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-3 x} \left (c_1 e^x-c_2 (x+1)\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), (x, 2)) + (5*x**3 - x**2)*Derivative(y(x), x) + (6*x**3 - 2*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False