 Hi friends, I am Purva and today we will discuss the following question. Find the equation of a curve passing through the point 0,2 given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5. Let us now begin with the solution. Now we know that the slope of the tangent to the curve at the point x,y is given by dy by dx. So let the x,y be a point on the curve. Therefore by the given conditions we have dy by dx is equal to x plus y minus 5 and this implies dy by dx minus y is equal to x minus 5. Let us mark this as equation 1. Now equation 1 is the linear differential equation of the form dy by dx plus py is equal to q. Therefore the integrating factor of this equation is given by e raised to the power integral p dx and we get this is equal to e raised to the power integral. Now here p is equal to minus 1 so we have minus 1 dx and this is further equal to e raised to the power. Now integral of minus 1 with respect to x is minus x. So we get the integrating factor as e raised to the power minus x. Now multiplying both the sides of equation 1 by e raised to the power minus x we get e raised to the power minus x into dy by dx minus y into e raised to the power minus x is equal to x into e raised to the power minus x minus 5 into e raised to the power minus x. And this implies now we can write e raised to the power minus x into dy by dx minus y into e raised to the power minus x as d upon dx of y into e raised to the power minus x. differentiating y into e raised to the power minus x with respect to x we will get e raised to the power minus x into dy by dx minus y into e raised to the power minus x and this is equal to x into e raised to the power minus x minus 5 into e raised to the power minus x. Now integrating both the sides with respect to x we get integral d upon dx of y into e raised to the power minus x dx is equal to integral x into e raised to the power minus x dx minus integral 5 into e raised to the power minus x dx and this implies now integrating left hand side we get y into e raised to the power minus x this is equal to now here we take x as the first term and e raised to the power minus x as the second term. So integrating x into e raised to the power minus x by Bipart's method we get x into integral of e raised to the power minus x dx minus integral derivative of x with respect to x into integral e raised to the power minus x dx dx minus now integrating 5 into e raised to the power minus x we get 5 into minus e raised to the power minus x plus c and this implies y into e raised to the power minus x is equal to x into now integral of e raised to the power minus x is minus e raised to the power minus x minus now differentiating x with respect to x gives one and integral of e raised to the power minus x is minus e raised to the power minus x so we get minus integral of minus e raised to the power minus x dx plus 5 into e raised to the power minus x plus c and this further implies y into e raised to the power minus x is equal to minus x into e raised to the power minus x minus e raised to the power minus x plus 5 into e raised to the power minus x plus c and this implies y into e raised to the power minus x is equal to minus x into e raise to the power minus x plus 4 into e raise to the power minus x plus c and the supplies now dividing both the sides by e raise to the power minus x we get y is equal to minus x plus 4 plus c into e raise to the power x and we mark this as equation 2 now in the question we are given that the curve passes through the point 0 comma 2 so as the curve passes through the point 0 comma 2 therefore we have putting x as 0 and y as 2 in equation 2 we get 2 is equal to 0 plus 4 plus c into e raise to the power 0 and this implies 2 is equal to 4 plus c because e raise to the power 0 is equal to 1 and this further implies minus 2 is equal to c now substituting the value of c in equation 2 we get y is equal to minus x plus 4 minus 2 into e raise to the power x and this is the required equation of the curve hence the required answer is y is equal to 4 minus x minus 2 into e raise to the power x this is our answer hope you have understood the solution bye and take care