If the solution curve of the differential equation (2x - 10 y ^ 3...

Given that: $\left(2x - 10{y}^{3}\right)dy + ydx = 0$

$⇒ \dfrac{dx}{dy} = 10{y}^{2} - \dfrac{2x}{y}$

$⇒ \dfrac{dx}{dy} + \dfrac{2x}{y} = 10{y}^{2}$

$⇒ \text{I.F.} = {e}^{2 ∫ \dfrac{1}{y}dx} = {y}^{2}$

$⇒ x ⋅ {y}^{2} = ∫ 10{y}^{4}dy + C$

$⇒ x ⋅ {y}^{2} = 2{y}^{5} + c$

Putx = 0 \text{ and } y = 1 ⇒ c = - 2

$⇒ x ⋅ {y}^{2} = 2{y}^{5} - 2$

Passing Through $( 2, \beta )$
$⇒ 2 ⋅ {\beta}^{2} = 2{\beta}^{5} - 2$

$⇒ {\beta}^{5} - {\beta}^{2} - 1 = 0$
root of the equations
${y}^{5} - {y}^{2} - 1 = 0$