44.3.31 problem 41

Internal problem ID [7012]
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 : 41
Date solved : Wednesday, March 05, 2025 at 04:02:20 AM
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 (1\right )&=4\\ y^{\prime }\left (1\right )&=-2 \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 24

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)-3*y(x) = 6*x+4; 
ic:=y(1) = 4, D(y)(1) = -2; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {9 \,{\mathrm e}^{1-x}}{2}+\frac {3 \,{\mathrm e}^{3 x -3}}{2}-2 x \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 31

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

\[ y(x)\to -2 x+\frac {9 e^{1-x}}{2}+\frac {3}{2} e^{3 x-3} \]

✓ Sympy. Time used: 0.225 (sec). Leaf size: 27

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(1): 4, Subs(Derivative(y(x), x), x, 1): -2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - 2 x + \frac {3 e^{3 x}}{2 e^{3}} + \frac {9 e e^{- x}}{2} \]