Exercises for differential equations 1.- Check the order, separability or linearity and homogeneity of the following differential equations: a) y' (t) = (y(t))2 b) (t + 1 ) (y' (t) ) = y(t) c) y"(t) + y'(t) - y(t) = 5 d) y' (t) - ty(t) =y"(t) e) y' (t) = -2 ty(t) f) y(t)= 2 y'(t)-5 t 2.- Obtain the analytic solutions of the following differential equations: a) y' (t) = (y(t))2 b) (t + 1) y (t) = y(t) c) y' (t) + sin (t) - y(t) = cos(t) d) y'(t) - ty(t) = 1 e) y' (t) =-2ty(t) 3.- Obtain the analytic solutions of the following systems of differential equations: a) x'(t) = 3 x(t) + 2 y(t) y' (t) = 2 x(t) + y(t) b) x' (t) = 2 x(t) + 0.5 y(t) y' (t) = - 0.5 x(t) + y(t) 4.- a) Prove that the differential equation y'"'- y"+ y'- y = 0, has a solution of type y(t)= et. b) Prove that the differential equation y'"ty" ty' ty =0, has a solution of type y(t)= et