What will be the general solution of differential equation for 2^x – y dx + 2^y – x dy = 0?

(a) 2^2x + 2^2y = c

(b) 2(2^2x + 2^2y) = c

(c) 3(2^2x + 2^2y) = c

(d) 4(2^2x + 2^2y) = c

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My question is from Linear First Order Differential Equations in portion Differential Equations of Mathematics – Class 12

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Right answer is (a) 2^2x + 2^2y = c

To explain I would say: We have, 2^x – y dx + 2^y – x dy = 0

Or, (2^x/2^y)dx + (2^y/2^x)dy = 0

Integrating both sides,

∫2^2xdx + ∫2^2ydy = k

Taking k = c/2log2 as, k, c and 2log2 are constant.

Or, 2^2x/2log2 + 2^2y/2log2 = c/2log2

Or, 2^2x + 2^2y = c