Question

Show that the given equation is not exact and an integrating factor depending on x alone or y alone does not exist. If possible, find an integrating factor in the form I(x,y) = xᵃ yᵇ , where a and b are suitably chosen, and use it to obtain the general solution.

(3xy+2y² )dx (3x²+4xy)dy=0

Answer 1

The given equation is not exact and there is no integrating factor in the form I(x,y) = xᵃyᵇ.

Step-by-step explanation:

To determine if the given equation is exact, we need to check if the partial derivatives of the coefficient functions with respect to x and y are equal. Let's check:

Partial derivative of (3xy+2y²) with respect to y: 4y

Partial derivative of (3x²+4xy) with respect to x: 6x+4y

Since the partial derivatives are not equal, the equation is not exact. Next, let's check if an integrating factor exists in the form I(x,y) = xᵃyᵇ. We can find the values of a and b by comparing the coefficients of dx and dy in the equation.

Coefficient of dx: 3xy+2y²

Coefficient of dy: 3x²+4xy

In order for an integrating factor to exist, we need the following condition to be satisfied: M_y - N_x = (2y-3x) - (6x+4y) = -9x-2y ≠ 0

Since -9x-2y is not equal to zero, an integrating factor in the form I(x,y) = xᵃyᵇ does not exist for this equation.

Answer 2

To show that the given differential equation is not exact, we compare the partial derivatives of M and N. We then find an integrating factor in the form I(x, y) = xʸ yʹ to make the equation exact. With the integrating factor, we multiply the entire equation by it and find the general solution by integrating Mdx + Ndy.

Step-by-step explanation:

The given differential equation (3xy+2y²)dx + (3x²+4xy)dy = 0 is not exact since ∂M/∂y ≠ ∂N/∂x, where M = 3xy + 2y² and N = 3x² + 4xy. To find an integrating factor of the form I(x, y) = xʸ yʹ which makes this equation exact, we search for constants a and b such that the equation becomes exact upon multiplying by this integrating factor.

Through the process of trial and error or systematic searching, we would find the values for a and b that make the equation exact. Once the integrating factor is found, we multiply the entire differential equation by this factor, which enables us to then find the potential function Φ(x, y) such that dΦ = Mdx + Ndy, leading us to the general solution of the differential equation.

If no such integrating factor exists, this would suggest that the differential equation must be approached by different methods for finding a solution.