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Question
Solve the following differential equation:
`"sec"^2 "x" * "tan y" "dx" + "sec"^2 "y" * "tan x" "dy" = 0`
Solution
`"sec"^2 "x" * "tan y" "dx" + "sec"^2 "y" * "tan x" "dy" = 0`
∴ `("sec"^2 "x")/"tan x" "dx" + ("sec"^2 "y")/"tan y" "dy" = 0`
Integrating both sides, we get
`int ("sec"^2"x")/"tan x" "dx" + int ("sec"^2 "y")/"tan y" "dy" = "c"_1`
Each of these integrals is of the type
`int ("f"'("x"))/("f"("x))` `"dx" = log |"f"("x")| + "c"`
∴ the general solution is
log |tan x| + log |tan y| = log c, where c1 = log c,
∴ `log |tan "x" * tan "y"| = log c`
∴ `tan "x" * tan"y" = "c"`
This is the general solution.
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