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.

Question

Philoid · Accepted Answer

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