8 30 points Consider the system of differential equations fo

8. (30 points) Consider the system of differential equations for the functions x(t) and y(t) x double dash - 3(x\')^2 + 4 sin x + e^t = 0 and y double dash + 3y\' - In y = sin(cos t). Convert this system to a first order system of differential equations in the form x\' = f(x) + g(t), x epsilon R^4.

Solution

Given:differential equation of

                 x(t)=x\'\'-3(x\')²+ 4sinx+e^t=0

                     if we apply integration on both sides

                      x\'-x³-4cosx+e^t=0

                       x\'=x³ +4cosx+e^t

        this is inthe form of: x\' = f(x)+g(t)

       and

              y(t)=y\'\'+3y\'-lny=sin(cost)

           integrate on both sides

             y\'+3y-(1/y)=-cos(cost)sint

             y\'=1/y-3y-cos(cost)sint

                =(1-1/3)y-cos(cost)sint

                =(2/3)y-cos(cost)sint

    this in in the form of y\'=f(y)+g(t).