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Consider a system of two nonlinear first-order ODEs, where $x$ and $y$ are functions of the independent variable $t$ : $x?=2tanh(x)?2xcos(y)+e_{x+3y}?1,y??=3cosh(x)?3e_{xy}+21?y+21?sin(x)$ (a) Write down in matrix form of the type $X?=AX$ with $X=(x,y)_{?}$ 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 $t??lim?x(t)$ and $t??lim?y(t)$.

(a) To linearize the system of ODEs around the point (x, y) = (0, 0), we ca