89.16.3 problem 3

Internal problem ID [24653]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 9. Nonhomogeneous Equations: Undetermined coefficients. Exercises at page 140
Problem number : 3
Date solved : Thursday, October 02, 2025 at 10:46:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }-3 y&=27 x^{2} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 26

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

\[ y = {\mathrm e}^{3 x} c_2 +{\mathrm e}^{-x} c_1 -9 x^{2}+12 x -14 \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 31

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

\begin{align*} y(x)&\to -9 x^2+12 x+c_1 e^{-x}+c_2 e^{3 x}-14 \end{align*}

✓ Sympy. Time used: 0.126 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-28*x**2 - 3*y(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{- x} + C_{2} e^{3 x} - \frac {28 x^{2}}{3} + \frac {112 x}{9} - \frac {392}{27} \]