Final answer:
To obtain the general solution to the equation y(dx/dy) - 8x = 3y⁶, we use separation of variables and solve the resulting linear first-order differential equation. The general solution is y = (Ce^(4x²) + (15/8)xe^(4x²))^(1/5), where C is the constant of integration.
Step-by-step explanation:
To obtain the general solution to the equation y(dx/dy) - 8x = 3y⁶, we can rearrange the equation and use separation of variables. We divide both sides by y⁶ and rewrite the equation as (1/y⁶)dy/dx - 8x/y⁶ = 3. Let's denote u = 1/y⁵, so du/dx = -5/y⁶ * dy/dx. Substituting these values, we get du/dx + 8xu = -15. This is now a linear first-order differential equation, and we can solve it using an integrating factor.
The integrating factor is e^(∫8xdx), which simplifies to e^(4x²). Multiplying both sides of the equation by this integrating factor gives us e^(4x²) * du/dx + 8x * e^(4x²) * u = -15 * e^(4x²). The left side of the equation is (u * e^(4x²))' (the derivative of the product), so we can rewrite the equation as (u * e^(4x²))' = -15 * e^(4x²). Integrating both sides gives us u * e^(4x²) = -15∫e^(4x²)dx + C, where C is the constant of integration.
Simplifying the integral and solving for u, we have u = Ce^(-4x²) - (15/8)xe^(-4x²). Now, substituting back the original value of u = 1/y⁵, we get 1/y⁵ = Ce^(-4x²) - (15/8)xe^(-4x²). Rearranging, we have y⁵ = 1/(Ce^(-4x²) - (15/8)xe^(-4x²)). Taking the reciprocal of both sides, we find the general solution to be y = (Ce^(4x²) + (15/8)xe^(4x²))^(1/5), where C is the constant of integration.