89.19.7 problem 7

Internal problem ID [24735]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 10. Nonhomogeneous Equations: Operational methods. Exercises at page 151
Problem number : 7
Date solved : Thursday, October 02, 2025 at 10:47:32 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }+9 y^{\prime \prime }+27 y^{\prime }+27 y&=15 x^{2} {\mathrm e}^{-3 x} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 24

ode:=diff(diff(diff(y(x),x),x),x)+9*diff(diff(y(x),x),x)+27*diff(y(x),x)+27*y(x) = 15*x^2*exp(-3*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-3 x} \left (\frac {1}{4} x^{5}+c_1 +c_2 \,x^{2}+c_3 x \right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 34

ode=D[y[x],{x,3}]+9*D[y[x],{x,2}]+27*D[y[x],{x,1}]+27*y[x]==15*x^2*Exp[-3*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{4} e^{-3 x} \left (x^5+4 c_3 x^2+4 c_2 x+4 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.254 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-15*x**2*exp(-3*x) + 27*y(x) + 27*Derivative(y(x), x) + 9*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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