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# Solution deferential equation dy/dx =cos(x+y)+sin(x+y), is

Arun
25763 Points
3 years ago
This is the solution.Integrate both sidesUsing variable and separable methodPut x+y = v1+dy/dx = dv/dxdy/dx = dv/dx – 1
Arun
25763 Points
3 years ago
DE : dy/dx = sin (x+y) + cos (x+y).
____________________________

Let : u = x + y.

∴ y = u - x

∴ dy/dx = (du/dx) - 1.
_____________________________

∴ the above DE now becomes :

... (du/dx) - 1 = sin u + cos u

∴ du/dx = ( 1 + cos u ) + sin u

∴ du/dx = ( 2 cos² u/2 ) + ( 2 sin u/2. cos u/2 )

∴ dividing both sides by ( 2 cos² u/2 ),

... (1/2) sec² u/2 (du/dx) = 1 + tan u/2

∴ ∫ [ (1/2) sec² u/2 / ( 1 + tan u/2 ) ] du = ∫ dx

∴ ∫ ( 1 / v ) dv = x, ...... v = 1 + tan u/2

∴ ln |v| = x + C

∴ ln | 1 + tan u/2 | = x + C

∴ ln | 1 + tan [(x+y)/2] | = x + C