A curve passes through the point (0, 2) and 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. Find the equation of the curve.
Formula :
i)
ii)
iii)
iv)
v) General solution :
For the differential equation in the form of
General solution is given by,
Where, integrating factor,
Answer :
The slope of the tangent to the curve
The sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at the given point by 5.
Therefore differential equation is
………eq(1)
Equation (1) is of the form
Where, and
Therefore, integrating factor is
………
General solution is
………eq(2)
Let,
Let, u=x-5 and v= e-x
………
………
………
Substituting I in eq(2),
Dividing above equation by e-x,
Therefore, general solution is
The curve passes through point (0,2) , therefore the above equation satisfies for x=0 and y=2,
Substituting c in general solution,
Therefore, equation of the curve is