The solution of the differential equation (x + y) dy = dx is: - Mathematics MCQ Mathematics Mock Test - 11 2026 | ExamTest.live

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The solution of the differential equation cos (x + y) d y = d x is:

Solution & Step-by-step Explanation

\frac{d y}{d x} = \frac{1}{c o s ( x + y )} = sec (x + y) .Let x + y = t . Differentiating w.r.t x : 1 + \frac{d y}{d x} = \frac{d t}{d x} ⟹ \frac{d y}{d x} = \frac{d t}{d x} - 1 .Substitute in the DE: \frac{d t}{d x} - 1 = sec t \frac{d t}{d x} = 1 + sec t = 1 + \frac{1}{c o s t} = \frac{1 + c o s t}{c o s t} \frac{c o s t}{1 + c o s t} d t = d x \int (1 - \frac{1}{1 + c o s t}) d t = \int d x \int (1 - \frac{1}{2 c o s ^{2} ( t /2 )}) d t = x + c \int (1 - \frac{1}{2} sec^{2} (t /2)) d t = x + c t - tan (t /2) = x + c Substitute t = x + y : (x + y) - tan (\frac{x + y}{2}) = x + c y = tan (\frac{x + y}{2}) + c .

Try it yourself before checking the explanation above.

The solution of the differential equation cos (x + y) d y = d x is:

y = x sec (y / x) + c

y + cos^{- 1} (y / x) = c

y = tan (\frac{x + y}{2}) + c

y = cot (\frac{x + y}{2}) + c

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