14.22.7 problem 7

Internal problem ID [2734]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.8, Systems of differential equations. The eigenva1ue-eigenvector method. Page 339
Problem number : 7
Date solved : Monday, January 27, 2025 at 06:12:29 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )+x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 2\\ x_{2} \left (0\right ) = 3 \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 33

dsolve([diff(x__1(t),t) = x__1(t)+x__2(t), diff(x__2(t),t) = 4*x__1(t)+x__2(t), x__1(0) = 2, x__2(0) = 3], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \frac {{\mathrm e}^{-t}}{4}+\frac {7 \,{\mathrm e}^{3 t}}{4} \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{-t}}{2}+\frac {7 \,{\mathrm e}^{3 t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 44

DSolve[{D[ x1[t],t]==1*x1[t]+1*x2[t],D[ x2[t],t]==4*x1[t]+1*x2[t]},{x1[0]==2,x2[0]==3},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{4} e^{-t} \left (7 e^{4 t}+1\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} \left (7 e^{4 t}-1\right ) \\ \end{align*}