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Find the particular solution of the following differential equation
`("d"y)/("d"x)` = e^{2y} cos x, when x = `pi/6`, y = 0.
Solution: The given D.E. is `("d"y)/("d"x)` = e^{2y} cos x
∴ `1/"e"^(2y) "d"y` = cos x dx
Integrating, we get
`int square "d"y` = cos x dx
∴ `("e"^(2y))/(2)` = sin x + c_{1}
∴ e^{–2y} = – 2sin x – 2c_{1}
∴ `square` = c, where c = – 2c_{1 }
This is general solution.
When x = `pi/6`, y = 0, we have
`"e"^0 + 2sin pi/6` = c
∴ c = `square`
∴ particular solution is `square`
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Solution
The given D.E. is `("d"y)/("d"x)` = e^{2y} cos x
∴ `1/"e"^(2y) "d"y` = cos x dx
Integrating, we get
`int bb("e"^(2y)) "d"y` = cos x dx
∴ `("e"^(2y))/(2)` = sin x + c_{1}
∴ e^{–2y} = – 2sin x – 2c_{1}
∴ e^{–2y} + 2sin x = c, where c = – 2c_{1 }
This is general solution.
When x = `pi/6`, y = 0, we have
`"e"^0 + 2sin pi/6` = c
∴ `1 + 2(1/2)` = c
∴ c = 2
∴ particular solution is e^{–2y} + 2sin x = 2
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