Reduce the following differential equation to the variable separable form and hence solve: dydxx + ydydx=cos(x + y) - Mathematics and Statistics

Sum

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

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

Solution

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.

Concept: Formation of Differential Equations
Is there an error in this question or solution?