Question

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

`cos^2 ("x - 2y") = 1 - 2 "dy"/"dx"`

Answer 1

`cos^2 ("x - 2y") = 1 - 2 "dy"/"dx"`      .....(1)

Put x - 2y = u. Then 1 - `2 "dy"/"dx" = "du"/"dx"`

∴ (1) becomes, `cos^2 "u" = "du"/"dx"`

∴ dx = `1/cos^2"u"`du

Integrating both sides, we get

∫ dx = ∫ sec²u du

∴ x = tan u + c

∴ x = tan (x - 2y) + c

This is the general solution.