Question

Solve the following differential equation:

`cos "x" * cos "y"  "dy" - sin "x" * "sin y" "dx" = 0`

Answer 1

`cos "x" * cos "y"  "dy" - sin "x" * "sin y" "dx" = 0`

`"cos y"/"sin y"  "dy" - "sin x"/"cos x"  "dx" = 0`

Integrating both sides, we get

`int cot"y"  "dy" - int "tan x"  "dx" = "c"_1`

∴ `log |sin "y"| - [- log |sec "x"|] = log "c",` where c₁ = log c

∴ `log |sin "y"| + log |sec "x"| = log "c"`

∴ `log |sin "y" * sec "x"| = log "c"`

∴ `sin "y" * cos "x" = "c"`

This is the general solution.