dD kgU Dnew dt dU kaDUnew dt Use Laplace transforms to sol

dD = kgU + Dnew dt dU =kaD+Unew dt Use Laplace transforms to solve this system for D(t) and U(t). From this solution determine: a) Who wins, b) How long the battle lasts, and c) Plot the number of destroyers and submarines during the course of the battle.

Solution

It is unclear how the equations for dD and DU are organized. Please explain the context and what the variables D and U stand for, so that a more detailed solution could be posted.

Here is an outline of an approach through Laplace transform;

The form of the equations looks like (with unknown functions X(t) and Y(t))

           X\' = aX+bY.....................(1)

           Y\' =cX+dY.......................(2)

with initial conditions X(0)=P and Y(0) =Q

Taking Laplace Transforms of (1) and (2) , we obtain

            sF(s)-P= aF(s) + bG(s).............(3)

            sG(s) -Q = cF(s) +dG(s)............(4)

This lead to two simultaneous equations for F(s) and G(s) :

           F(s) (s-a) -bG(s) =P...................(5)

           -cF(s) + (s-d) G(s) =Q...............(6)

Solving these, we obtain expressions for F(s) and G(s). By taking inverse transfroms , X(t) and Y(t) are obtained.

Depending upon the nature of he constants a,b,c,d , and taking the limit as t tends to infinity, we can definitive answers to (a),(b) and (c)