Determine the solution to the 1st order differential equatio

Determine the solution to the 1st order differential equation listed below. Using the initial condition, evaluation the constant of integration. y\' = y y(o) = 1

Solution

Given equation: y\'=y

                          y\'-y=0

we need to Find the general solution of this equation:

       First let’s rewrite this equation as: y\'-y=0

                                                               y\'=y

                                                                dy/dt=y

        

Then, assuming y 0, divide both sides by y:

                                                             dy/y=dt

       Now what we have here are two derivatives which are equal.

It implies that the anti derivatives of the two sides must differ only by a constant of integration. Integrate both sides:

                                                                ln|y|=t+C

                                                                 |y|=e^(t+c) =e^c.e^t = C₁ e^t.

           Here,

C₁ = e^c is an arbitrary, but always positive constant.

if take y(0)=1 then,

                                          dy/dt=1, it comes constant.