\(dy/dx= e^{4x}(y-3)\)

\(dy/dx= (e^{4x})(y-3)\)

\(dy/((y-3))= (e^{4x})dx\)

\(\int 1/(y-3) dy= \int (e^{4x})dx\)

\(\ln(y-3)= e^{4x}/4+c\)

Apply exponential on both sides

\(e^{\ln(y-3)}= e^{(e^4x)/4+c)}\)

Thus, the solution of the given first order differential equation is

\(y= e^{(e^{4x}/4+c)}+3\)