Solve the Differential Equation. Use the Fact that the given Equation is Homogeneous. dy/dx = (x² + 8y²) / 3xy

We will be solving this by using the first-order separable ordinary differential equation. 

Answer: The general solution of differential equation dy/dx = (x² + 8y²) / 3xy is y = ± √c₁x^16/3 - x²/5

Let's solve this step by step.

Explanation:

Given, differential equation: dy/dx = (x² + 8y²) / 3xy

Divide the numerator and denominator by x² in the RHS

dy/dx = (1 + 8y²/x²) / 3(y/x) --------------- (1)

Let, y/x = v ----------- (2)

⇒ y = xv

Differentiating with respect to 'x' we get,

⇒ dy/dx = v + x dv/dx ------------- (3)

Substitue the values of (2) and (3) in (1):

⇒ v + x dv/dx = (1 + 8v²) / 3v

⇒ 3v² + 3vx dv/dx = (1 + 8v²)

⇒ 3vx dv/dx = (1 + 5v²)

⇒ 3v dv / (1 + 5v²) = dx / x

Integrate on both sides:

∫ 3v dv /(1 + 5v²) = ∫ dx / x ---------------- (4)

Let z = 1 + 5v²

⇒ dz/dv = 10v

⇒ dv = dz/10v

Substitue the value of dv in (4)

∫ (3v/z) (dz/10v) = ∫ dx / x

∫ 3 dz /10z = ∫ dx / x

⇒ 3/10 ln(z) = ln (xc)

⇒ ln(z) = 10/3 ln(xc)

Substitue z = 1 + 5v²

ln(1 + 5v²) =  ln(xc)^10/3

ln(1 + 5(y/x)²) =  ln(xc)^10/3 [Since, v = y / x]

⇒ 1 + 5(y/x)² =  (xc)^10/3

⇒ (y/x)² = (1/5)cx^10/3 - 1/5

⇒ y² = (1/5)cx^10/3 . x² - x²/5

⇒ y² = (1/5)cx^16/3 - x²/5

⇒ y = ± √(1/5)cx^16/3 - x²/5

⇒ y = ± √c₁x^16/3 - x²/5

Thus, the general solution of differential equation dy/dx = (x² + 8y²) / 3xy is y = ± √c₁x^16/3 - x²/5