Exercise 3. [15 pts] superposition principle for nth order linear, homogeneous ode we have done the n-d, 1st order linear, homogeneous ode in class. since nth order linear, homogeneous ode can always to rewritten into n-d, 1st order linear, homogeneous ode, we already know the principle holds. here we prove the principle from a dierent route: by direct using the linearity of dierentiation. (a) (3 pts) from the denition of derivative, df dt = lim h→0 f(t + h) − f(t) h , show the linearity of the dierentiation (assuming all derivatives exist), d dt (f + g) = df dt + dg dt (b) (5 pts) using the results in (a), argue that the same holds for the nth derivative (assuming the nth derivative of f and g both exist) d n dt n (f + g) = d nf dt n + d ng dt n