exact solution to differential equation

Question

y'=y , y(2)=3

Answer 1

dy/dx=y

 

dy/y=dx

 

integrating we will get

 

ln y = x +c

  y= e^x+c   =e^xe^c

 y = Ce^x    ; putting e^c =C 

Now we have written y as a  function of x and its given when x=2 ,y=3

 

So we will get value of C by putting initial condition  :   3=Ce²   => C=3/e² =3e^-2

 

Therefore exact solution will be : y = 3e^-2e^x

 

                                  or           y = 3e^x-2