38.3.29 problem 39

Internal problem ID [8296]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 1. Introduction to differential equations. Review problems at page 34
Problem number : 39
Date solved : Tuesday, September 30, 2025 at 05:21:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }-3 y&=6 x +4 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 20

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)-3*y(x) = 6*x+4; 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{3 x}}{2}-\frac {{\mathrm e}^{-x}}{2}-2 x \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 27

ode=D[y[x],{x,2}]-2*D[y[x],x]-3*y[x]==6*x+4; 
ic={y[0]==0,Derivative[1][y][0] == 0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.110 (sec). Leaf size: 19

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

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