Help asap please

Question

Find the orthogonal trajectories of the following family of curves 𝑥^ 2 + 3𝑦^ 2 = 𝑐𝑦

PLEASE WRITE THE SOLUTION WITH ALL STEPS

Answer 1

We differentiate implicitly: 2x + (6y- c)y' = 0; y' = 2x/(c - 6y).

For orthogonal traeyctory y' = - (c - 6y)/(2x); dy(6y - c) = dx/(2x)

Integration give us: 1/6ln(6y - c) = 1/2 lnx + 1/6lnC; 6y - c = Cx³

From given equation c = x²/y + 3y.

So, orthogonal trayectory is: 3y - x²/y = Cx³, or 3y² - x² = Cx³y