Question

Reduce the following differential equation to the variable separable form and hence solve:

`"dy"/"dx" = cos("x + y")`

Answer 1

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.