78.27.1 problem 5 (a,c)

Internal problem ID [18466]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 10. Systems of First Order Equations. Section 55. Linear systems. Problems at page 496
Problem number : 5 (a,c)
Date solved : Tuesday, January 28, 2025 at 11:50:29 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )+3 y\\ y^{\prime }&=3 x \left (t \right )+y \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 5\\ y \left (0\right ) = 1 \end{align*}

✓ Solution by Maple

Time used: 0.064 (sec). Leaf size: 33

dsolve([diff(x(t),t) = x(t)+3*y(t), diff(y(t),t) = 3*x(t)+y(t), x(0) = 5, y(0) = 1], singsol=all)
 

\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{4 t}+2 \,{\mathrm e}^{-2 t} \\ y &= 3 \,{\mathrm e}^{4 t}-2 \,{\mathrm e}^{-2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 38

DSolve[{D[x[t],t]==x[t]+3*y[t],D[y[t],t]==3*x[t]+y[t]},{x[0]==5,y[0]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{-2 t} \left (3 e^{6 t}+2\right ) \\ y(t)\to e^{-2 t} \left (3 e^{6 t}-2\right ) \\ \end{align*}