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Find the general solution of the differential equation dy/dx = sin(x + y) + cos(x + y).

1. log[1 + tan((x + y)/2)] + c = 0
2. log[1 + tan((x + y)/2)] = x + c
3. log[1 - tan((x + y)/2)] = x + c
4. None of these

Correct answer: log[1 + tan((x + y)/2)] = x + c

Solution

Let u = x+y. Then du/dx = 1 + dy/dx = 1 + sin u + cos u. Separating: du/(1+sin u+cos u) = dx. Using t = tan(u/2): 1+sin u+cos u = 2(1+t)/(1+t²) and du = 2dt/(1+t²). So the integral becomes integral dt/(1+t) = integral dx => ln|1+t| = x+c => log[1+tan((x+y)/2)] = x+c.

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