Question

dy/dx = sec(x+y) solve the differential equation..​

Answer 1

Answer:

tan{(x + y)/2} = x + c

Step-by-step explanation:

The given differential equation is

dy/dx = sec(x + y) ..... (1)

Let x + y = z. Then differentiating with respect to x, we gey

1 + dy/dx = dz/dx

From (1), we get

dz/dx - 1 = secz

or, dz/dx = 1 + secz

or, dz / (1 + cosz) = dx

or, (1 - cosz) dz / (1 - cos²z) = dx

or, (1 - cosz) dz / sin²z = dx

or, cosec²z dz - cosecz cotz dz = dx

Integrating, we get

∫ cosec²z dz - ∫ cosecz cotz dz = ∫ dx

or, - cotz + cosecz = x + c ,

where c is constant of integration

or, cosecz - cotz = x + c

or, 1/sinz - cosz/sinz = x + c

or, (1 - cosz)/sinz = x + c

or, {2 sin²(z/2)}/{2 sin(z/2) cos(z/2)} = x + c

or, tan(z/2) = x + c

or, tan{(x + y)/2} = x + c

which is the required integral.