Help me solve question

Consider the following equation with constant _. ut = (f(ux))x _ _uxxxx, f(v) = v2 _ v A. Linearize this equation around u = 0 and find the principal mode solution of the form e_t ikx. For which values of _ are there unstable modes, i.e., modes with _ = 0 for real k? For these values, find the maximally unstable mode, i.e., the value of k with the largest positive value of _. B. Consider the steady solution of the (fully nonlinear) problem. Show that the resulting equation can be written as a second order autonomous ODE for v = ux and draw the corresponding phase plane.

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