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sec^(2)x tany dx + sec^(2)y tan x dy = d...

` sec^(2)x tany dx + sec^(2)y tan x dy = dy =0`

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`sec^(2) tan y dx + sec^(2) y. tan x dy =0`
` (sec^(2)x)/(tanx) dx + (sec^(2)y)/(tan y) dy =0`
Integrating , we get ,
` int (sec^(2)y)/(tanx) dx + int(sec^(2)y)/(tany) 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 " c_(1) = log c `
` log|tan x. tany | = log c`
tan x. tan y = c
This is the general solution.
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Knowledge Check

  • Find the particular solution of the following : sec^(2)x tan y dx - sec^(2)y tan x dy = 0 , given that y= pi/6, x= pi/3 .

    A
    `tan x = 2 tan y`
    B
    `tan x =tan y`
    C
    `tan x = 3 tan y`
    D
    `tan x = 5 tan y`

  • Solution of the differential equation tan y.sec^(2) x dx + tan x. sec^(2)y dy = 0 is

    A
    tan x + tan y = k
    B
    tan x - tan y = k
    C
    `(tan x)/(tan y)=k`
    D
    tanx.tany = k

  • The solution of the differential equation sec^2x tan y dx+sec^2y tanx dy=0 is :

    A
    tan y tan x = c
    B
    `(tan y)/(tan x)=c`
    C
    `(tan^2x)/(tany)=c`
    D
    none of these

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