A solution of `y=2x (dy/dx) + x^2(dy/dx)^4` is

Options

A. `y = 2sqrt(C ) x^(1//4)+C`
B. `y= 2 sqrt( C) x^(2) - C^(2)`
C. ` y = 3 sqrt(C ) (x+3)`
D. ` y = 2 sqrt(Cx)+C^(2)`

Answer

d

Explanation

Given , ` y = 2x ((dy)/(dx))+ x^(2)((dy)/(dx))^(4)`
Let `t = (dt)/(dx)` and then differentiating both sides w.r.t.x, we get
`t = 2t + 2x (dt)/(dx) + 2xt^(4) +4t^(2) +x^(2) (dt)/(dx)`
` rArr 0 = t (1+2xt^(3)) +2x (dt)/(dx) (1+2t^(3)x)`
` rArr (t+2x (dt)/(dx))(1+2t^(3)x)=0`
`rArr t+2x(dt)/(dx) = 0 rArr 2(dt)/t = - (dx)/x`
` rArr 2 log t + log x = log C`
` rArr t^(2)x = C rArr (dy)/(dx) = sqrt(C/x)`
On integrating this equation , we get
` y = 2 sqrt(Cx) +C^(2)`