14.20.5 problem 5

Internal problem ID [2702]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.14, The method of elimination for systems. Excercises page 258
Problem number : 5
Date solved : Monday, January 27, 2025 at 06:12:01 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = 2\\ y \left (0\right ) = 3 \end{align*}

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 33

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

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 44

DSolve[{D[x[t],t]==x[t]+y[t],D[y[t],t]==4*x[t]+y[t]},{x[0]==2,y[0]==3},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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