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The substitution $y = z^{α}$ transforms the differential equation $(x^{2} y^{2} - 1) dy + 2 {xy}^{3} dx = 0$ into a homogeneous differential equation for:

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$α = - 1$

$α = 1$

no value of $α$

answer is A.

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Detailed Solution

$(x^{2} z^{2 α} - 1) {αz}^{α - 1} dz + 2 {xz}^{3 α} dx = 0$

Or $α (x^{2} z^{3 α - 1} - z^{α - 1}) dz + 2 {xz}^{3 α} dx = 0$

For homogeneous every term must be of the same degree

$3 α + 1 = α - 1 \Rightarrow α = - 1$

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