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 : Tuesday, February 04, 2025 at 12:47:17 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*}

✓ Solution by Maple

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

dsolve([diff(x(t),t)=-lambda__1*x(t),diff(y(t),t)=lambda__1*x(t)-lambda__2*y(t)],singsol=all)
 

\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*}

✓ Solution by Mathematica

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

DSolve[{D[x[t],t]==-L1*x[t],D[y[t],t]==L1*x[t]-L2*y[t]},{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*}