Question

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

• A. $\alpha =-1$
• B. $0$
• C. $\alpha =1$
• D. no value for $\alpha$

Answer 1

The correct option is A $\alpha =-1$

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