(Solved): Consider a system of two nonlinear first-order ODEs, where \( x \) and \( y \) are functions of the ...

Consider a system of two nonlinear first-order ODEs, where \( x \) and \( y \) are functions of the independent variable \( t \) : \[ \dot{x}=2 \tanh (x)-2 x \cos (y)+e^{x+3 y}-1, \quad \dot{y}=3 \cosh (x)-3 e^{x y}+\frac{1}{2} y+\frac{1}{2} \sin (x) . \] (a) Write down in matrix form of the type \( \mathbf{X}=A \mathbf{X} \) with \( \mathbf{X}=(x, y)^{\top} \) the system obtained by linearisation of the above equations around the point \( x=y=0 \). Specify the elements of the matrix \( A \). (b) Find the eigenvalues and eigenvectors of the matrix \( A \) obtained in (a). Write down the general solution of the linear system. (c) What type of fixed point is the equilibrium solution \( x=y=0 \) ? Sketch the phase portrait of the linear system. (d) Find the solution of the linear system corresponding to the initial conditions \( x(0)=1, y(0)=0 \). Determine the values \( \lim _{t \rightarrow \infty} x(t) \) and \( \lim _{t \rightarrow \infty} y(t) \).