89.16.8 problem 8

Internal problem ID [24658]
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 : 8
Date solved : Thursday, October 02, 2025 at 10:46:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime }-2 y&=12 \,{\mathrm e}^{-2 x} \end{align*}

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

ode:=diff(diff(y(x),x),x)+diff(y(x),x)-2*y(x) = 12*exp(-2*x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 32

ode=D[y[x],{x,2}]+D[y[x],{x,1}]-2*y[x]==12*Exp[-2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{3} e^{-2 x} \left (-12 x+3 c_2 e^{3 x}-4+3 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.117 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*y(x) + Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 12*exp(-2*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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