14.20.8 problem 8

Internal problem ID [2705]
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 : 8
Date solved : Monday, January 27, 2025 at 06:12:04 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 37

dsolve([diff(x(t),t) = 3*x(t)-2*y(t), diff(y(t),t) = 4*x(t)-y(t), x(0) = 1, y(0) = 5], singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \left (-4 \sin \left (2 t \right )+\cos \left (2 t \right )\right ) \\ y &= {\mathrm e}^{t} \left (-3 \sin \left (2 t \right )+5 \cos \left (2 t \right )\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 40

DSolve[{D[x[t],t]==3*x[t]-2*y[t],D[y[t],t]==4*x[t]-y[t]},{x[0]==1,y[0]==5},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^t (\cos (2 t)-4 \sin (2 t)) \\ y(t)\to e^t (5 \cos (2 t)-3 \sin (2 t)) \\ \end{align*}