89.22.16 problem 16

Internal problem ID [24804]
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 161
Problem number : 16
Date solved : Thursday, October 02, 2025 at 10:48:09 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }+3 y^{\prime \prime }-4 y&=16 x^{3}+20 x^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 33

ode:=diff(diff(diff(y(x),x),x),x)+3*diff(diff(y(x),x),x)-4*y(x) = 16*x^3+20*x^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {27}{2}+\left (c_3 x +c_2 \right ) {\mathrm e}^{-2 x}-4 x^{3}-5 x^{2}+c_1 \,{\mathrm e}^{x}-18 x \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 45

ode=D[y[x],{x,3}]+3*D[y[x],{x,2}]-4*y[x]==16*x^3+20*x^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -4 x^3-5 x^2+x \left (-18+c_2 e^{-2 x}\right )+c_1 e^{-2 x}+c_3 e^x-\frac {27}{2} \end{align*}

✓ Sympy. Time used: 0.079 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-16*x**3 - 20*x**2 - 4*y(x) + 3*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{3} e^{x} - 4 x^{3} - 5 x^{2} - 18 x + \left (C_{1} + C_{2} x\right ) e^{- 2 x} - \frac {27}{2} \]