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  4. The substitution y=zα transforms the differential equation (x2 y2-1) d y+2 x y3 d x=0 into a homogeneous differential equation for

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Q. The substitution $y=z^{\alpha}$ transforms the differential equation $\left(x^2 y^2-1\right) d y+2 x y^3 d x=0$ into a homogeneous differential equation for

Differential Equations

Solution:

$\left(x^2 z^{2 \alpha}-1\right) \alpha z^{\alpha-1} d z+2 x z^{3 \alpha} d x=0 $
or $\alpha\left(x^2 z^{3 \alpha-1}-z^{\alpha-1}\right) d z+2 x z^{3 \alpha} d x=0$
for homogeneous every term must be of the same degree, $3 \alpha+1=\alpha-1 \Rightarrow \alpha=-1 \Rightarrow A$