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

Question

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 (A) α=-1 (B) 0 (C) α=1 (D) no value of α.

Tardigrade Admin · Accepted Answer

(x2 z2 α-1) α zα-1 d z+2 x z3 α d x=0 or α(x2 z3 α-1-zα-1) d z+2 x z3 α d x=0 for homogeneous every term must be of the same degree, 3 α+1=α-1 ⇒ α=-1 ⇒ A

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

$α = - 1$

0

$α = 1$

no value of $α$ .

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

Q. The substitution y=zα transforms the differential equation (x2y2−1)dy+2xy3dx=0 into a homogeneous differential equation for

α=−1

0

α=1

no value of α.

(x2z2α−1)αzα−1dz+2xz3αdx=0 or α(x2z3α−1−zα−1)dz+2xz3αdx=0 for homogeneous every term must be of the same degree, 3α+1=α−1⇒α=−1⇒A

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

$α = - 1$

$α = 1$

no value of $α$ .

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