Find the general solution to each of the following odes. then, decide whether or not the set of solutions form a vector space. explain your reasoning. compare your answers to the previous problem. recall that the general solution has the form y(t) = yh(t) + yp(t).

(a) y' - 2y = 0
(b) y' - 2y = 1
(c) y" - 4y = 0
(d) y" - 4y = e^(3t)

(A) with .

(B) with

(C) with

(D) with ,

Step-by-step explanation

(A) We can see this as separation of variables or just a linear ODE of first grade, then . With this answer we see that the set of solutions of the ODE form a vector space over, where vectors are of the form with real.

(B) Proceeding and the previous item, we obtain . Which is not a vector space with the usual operations (this is because ), in other words, if you sum two solutions you don't obtain a solution.

(C) This is a linear ODE of second grade, then if we set and we obtain the characteristic equation and then the general solution is with , and as in the first items the set of solutions form a vector space.

(D) Using C, let be we obtain that it must satisfies and then the general solution is with , and as in (B) the set of solutions does not form a vector space (same reason! as in (B)).  

step-by-step explanation: