The equation of the curve passing through the point (0,2) given that the sum of the ordinate and abscissa of any point exceeds the slope of the tangent to the curve at that point by 5 is

Step-by-step solution

The curve y = f(x) satisfies that for any point (x, y) , the sum of the abscissa and ordinate exceeds the slope of the tangent by 5: y + x = dy dx + 5 . Rearranging gives the linear differential equation dy dx - y = x - 5 . With P(x) = -1 and Q(x) = x - 5 , the integrating factor is e^ -x . The general solution is y e^ -x = (x - 5)e^ -x , dx + C . Using integration by parts, (x - 5)e^ -x , dx = e^ -x (4 - x) , yielding y e^ -x = e^ -x (4 - x) + C . Dividing by e^ -x gives y = 4 - x + Ce^x . Applying the initial con