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Question
Reduce the following differential equation to the variable separable form and hence solve:
`"dy"/"dx" = cos("x + y")`
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Solution
Given, equation is `"dy"/"dx"` = cos (x + y)
Put x + y = u,
Then `1 + "dy"/"dx" = "du"/"dx"`
∴ `"dy"/"dx" = "du"/"dx" - 1`
∴ (1) becomes, `"du"/"dx" - 1` = cos u
∴ `"du"/"dx"` = 1 + cos u
∴ `1/(1 + cos "u")`du = dx
Integrating both sides, we get
`int 1/(1 + cos "u") "du" = int "dx"`
∴ `int 1/(2cos^2 ("u"/2)) "du" = int "dx"`
∴ `1/2 int sec^2 ("u"/2)"du" = int "dx"`
∴ `1/2 (tan("u"/2))/(1/2) = "x" + c`
∴ `tan (("x + y")/2)` = x + c
This is the general solution.
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