Consider a system of two nonlinear first-order ODEs, where x and y are functi ...
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)+ex+3y?1,y??=3cosh(x)?3exy+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).