The solution of the differential equation d y d x -y=1 given y(0)=1 , is

Step-by-step solution

The given differential equation is a linear differential equation of the form dy dx + Py = Q , where P = -1 and Q = 1 . The integrating factor (IF) is given by IF = e^ P dx = e^ -1 dx = e^ -x . Multiplying both sides of the differential equation by the integrating factor, we get e^ -x dy dx - e^ -x y = e^ -x , which can be written as d dx (y e^ -x ) = e^ -x . Integrating both sides with respect to x , we have y e^ -x = e^ -x dx = -e^ -x + C . Multiplying by e^x , we obtain y = -1 + C e^x . Using the initial conditi