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Question
Reduce the following differential equation to the variable separable form and hence solve:
`"x + y""dy"/"dx" = sec("x"^2 + "y"^2)`
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Solution
`"x + y""dy"/"dx" = sec("x"^2 + "y"^2)` ....(1)
Put x2 + y2 = u
∴ 2x + 2y`"dy"/"dx" = "du"/"dx"`
∴ x + y`"dy"/"dx" = 1/2 * "du"/"dx"`
∴ (1) becomes, `1/2 * "du"/"dx" = sec"u"`
∴ `1/(sec "u") = 2 * "dx"`
Integrating both sides, we get
∫ cos u du = 2 ∫ dx
∴ sin u = 2x + c
∴ sin (x2 + y2) = 2x + c
This is the general solution.
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