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.

Explanation

Let any point on the curve be `(x,y)`.
Given that, `(dy)/(dx)=x+y-5`
`implies (dy)/(dx)-y=x-5`
Here , `P=-1`, `Q=x-5`
`:. I.F.=e^(int-1dx)=e^(-x)`
and general solution
`ye^(-x)=int(x-5)e^(-x)dx+c`
`=-(x-5)e^(-x)+int1*e^(-x)dx+c`
`=-(x-5)e^(-x)-e^(-x)+c`
`implies y=4-x+c*e^(x)`
This curve passes through `(0,2)`
`2=4-0+cimpliesc=-2`
`:. y=4-x-2e^(x)`
`implies y+x+2e^(x)=4`