44.6.54 problem 59

Internal problem ID [7198]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.3 Linear equations. Exercises 2.3 at page 63
Problem number : 59
Date solved : Wednesday, March 05, 2025 at 04:20:52 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-\lambda _{1} x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=\lambda _{1} x \left (t \right )-\lambda _{2} y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.312 (sec). Leaf size: 42

ode:=[diff(x(t),t) = -lambda__1*x(t), diff(y(t),t) = lambda__1*x(t)-lambda__2*y(t)]; 
dsolve(ode);
 

\begin{align*} x &= c_2 \,{\mathrm e}^{-\lambda _{1} t} \\ y &= \left (-\frac {c_2 \lambda _{1} {\mathrm e}^{-\lambda _{1} t +\lambda _{2} t}}{-\lambda _{2} +\lambda _{1}}+c_1 \right ) {\mathrm e}^{-\lambda _{2} t} \\ \end{align*}

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 63

ode={D[x[t],t]==-L1*x[t],D[y[t],t]==L1*x[t]-L2*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to c_1 e^{-\text {L1} t} \\ y(t)\to \frac {e^{-(t (\text {L1}+\text {L2}))} \left (e^{\text {L1} t} ((c_1+c_2) \text {L1}-c_2 \text {L2})-c_1 \text {L1} e^{\text {L2} t}\right )}{\text {L1}-\text {L2}} \\ \end{align*}

✓ Sympy. Time used: 0.224 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
lambda__1 = symbols("lambda__1") 
lambda__2 = symbols("lambda__2") 
x = Function("x") 
y = Function("y") 
ode=[Eq(lambda__1*x(t) + Derivative(x(t), t),0),Eq(-lambda__1*x(t) + lambda__2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {C_{1} \left (\lambda ^{1} - \lambda ^{2}\right ) e^{- \lambda ^{1} t}}{\lambda ^{1}}, \ y{\left (t \right )} = C_{1} e^{- \lambda ^{1} t} + C_{2} e^{- \lambda ^{2} t}\right ] \]