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

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Consider a system of two nonlinear first-order ODEs, where $$x$$ and $$y$$ are functions of the independent variable $$t$$ : $$\stackrel{?}{x}=2\tanh (x)?2x\cos (y)+e^{x+3y}?1,\stackrel{?}{y}=3\cosh (x)?3e^{xy}+\frac{1}{2}y+\frac{1}{2}\sin (x)$$ (a) Write down in matrix form of the type $$\stackrel{?}{\mathbf{X}}=A\mathbf{X}$$ with $$\mathbf{X}=(x,y{)}^{?}$$ 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 $$\underset{t??}{\lim }x(t)$$ and $$\underset{t??}{\lim }y(t)$$.