Question

For the differential equation, find the general solution:

`(x + y) dy/dx = 1`

Answer 1

Differential equations,

`(x + y) dy/dx = 1`

`therefore dx/dy = x + y`

or `dx/dy - x = y`

Comparing with the differential equation, `dx/dy + Px = Q`,

P = -1, Q = y

`I.F. = e^(int P dx) = e^(int (- 1)dy) = e^(- y)`

The solution of the differential equation is:

`x × I.F. = int Q xx I.F. dy + C`

`=> x xx e^(- y) = int y * e^(- y) dy + C`

On integrating piecewise,

`xe^(- y) = y (e^(- y)/(-1)) - int 1((e^(- y))/(-1)) dy + C`

`= - ye^(- y) + e^(-y)/(- 1) dy + C`

`= - ye^-y - e^(- y) + C`

or x = - y - 1 + Ce^y

∴ x + y + 1 = Ce^y

This is the desired solution.