# Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0. - Mathematics and Statistics

Sum

Solve the differential equation cos(x +y) dy = dx hence find the particular solution for x = 0 and y = 0.

#### Solution

Cos ( x + y )dy = dx
∴ dy/dx = 1/[ cos ( x + y )]
Let x + y = t
∴ 1 + dy/dx = dt/dx

∴ dt/dx - 1 = 1/[ cos t ]

dt/dx = 1/[cost] + 1

dt/dx = [ 1 + cost ]/cost

∴ cost/[ 1 + cost ]dt = dx
Integrating both side.
∴ int cost/[ 1 + cost ]dt = int dx

∴ int [ cost( 1 - cost )]/sin^2t dt = x + c

∴ int (cosect.cot t - cot^2 t) dt = x + c

∴ int ( cosec t.cot t - cosec^2 t + 1 )dt = x + c

∴ - cosect + cot t + t = x + c
∴  [cos t]/[sin t] - 1/[sin t] + t = x + c

- tan[( x + y )/2] + x + y = x + c

∴ -tan[( x + y )/2]+ y = c

Putting x = 0, y = 0
∴ -tan[( 0 + 0 )/2]+ 0 = c

∴ c = 0

∴ y = tan[( x + y )/2]

Concept: General and Particular Solutions of a Differential Equation
Is there an error in this question or solution?