89.16.4 problem 4

Internal problem ID [24654]
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 : 4
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&=-6 x^{2}-8 x +4 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 22

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)-3*y(x) = -6*x^2-8*x+4; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 27

ode=D[y[x],{x,2}]-2*D[y[x],{x,1}]-3*y[x]==4-8*x-6*x^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.118 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(6*x**2 + 8*x - 3*y(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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