50.29.2 problem 2(b)

Internal problem ID [8198]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 10. Systems of First-Order Equations. Section A. Drill exercises. Page 400
Problem number : 2(b)
Date solved : Monday, January 27, 2025 at 03:45:42 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )\\ y^{\prime }\left (t \right )&=4 x \left (t \right )+y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 35

dsolve([diff(x(t),t)=x(t)+y(t),diff(y(t),t)=4*x(t)+y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{-t} \\ y &= 2 c_{1} {\mathrm e}^{3 t}-2 c_{2} {\mathrm e}^{-t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 70

DSolve[{D[x[t],t]==x[t]+y[t],D[y[t],t]==4*x[t]+y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{4} e^{-t} \left (2 c_1 \left (e^{4 t}+1\right )+c_2 \left (e^{4 t}-1\right )\right ) \\ y(t)\to \frac {1}{2} e^{-t} \left (2 c_1 \left (e^{4 t}-1\right )+c_2 \left (e^{4 t}+1\right )\right ) \\ \end{align*}