Question

Find an integrating factor for the following differential equation. Use the notation mu(x) or mu(y) for the function μ(x) or μ(y), respectively:

(6xy² + 2y)dx + (12x²y + 6x + 3)dy = 0.

Answer 1

To find an integrating factor for the given differential equation (6xy² + 2y)dx + (12x²y + 6x + 3)dy = 0, we can use the formula μ(x) = e^∫P(x)dx, where P(x) is the coefficient of dx in the equation.

Step-by-step explanation:

To find an integrating factor for the given differential equation (6xy² + 2y)dx + (12x²y + 6x + 3)dy = 0, we can use the formula μ(x) = e∫P(x)dx, where P(x) is the coefficient of dx in the equation.

Here, P(x) = 6xy² + 2y. So, we need to find the integral of this expression with respect to x.

Integrating 6xy² + 2y with respect to x gives us 3x²y² + 2xy + g(y), where g(y) is a function of y only. Therefore, the integrating factor μ(x) = e3x²y² + 2xy + g(y).