Q4: A two-dimensional fluid flow is characterized by Eulerian description as u = ax + t^2, v = by - t^2, (a + b) = 0. Derive the corresponding Lagrangian description (given that the initial position of the fluid particle at t0 = 0 is x0, y0)? What is the acceleration in x and y for these two descriptions?
Note: the general solution for a first-order, inhomogeneous, linear ODE dy/dx + p(x)y = Q(x) reads y = e^{-∫ P(x)dx} (∫ [Q(x)e^{∫ P(x)dx}]dx + C)
Where C is an arbitrary constant. And, ∫ x^2e^{mx}dx = (x^2/m - 2x/m^2 + 2/m^3)e^{mx}