14.20.9 problem 9

Internal problem ID [2706]
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 : 9
Date solved : Monday, January 27, 2025 at 06:12:05 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.090 (sec). Leaf size: 33

dsolve([diff(x(t),t) = 4*x(t)+5*y(t)+4*exp(t)*cos(t), diff(y(t),t) = -2*x(t)-2*y(t), x(0) = 0, y(0) = 0], singsol=all)
 

\begin{align*} x \left (t \right ) &= -\frac {{\mathrm e}^{t} \left (-4 \cos \left (t \right ) t -12 \sin \left (t \right ) t -4 \sin \left (t \right )\right )}{2} \\ y &= -4 \sin \left (t \right ) {\mathrm e}^{t} t \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 33

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

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