To check for exactness, we take the partial derivative of x²y y⁴ with respect to y and the partial derivative of 2x³y⁴ with respect to x.

∂/∂y (x²y y⁴) = x²(4y³) = 4x²y³

∂/∂x (2x³y⁴) = 6x²y⁴

Since the partial derivative of x²y y⁴ with respect to y is equal to the partial derivative of 2x³y⁴ with respect to x, the equation is exact.


To find the potential function, we integrate the first term with respect to x and integrate the second term with respect to y.

∫x²y y⁴ dx = x³y⁴/3 + c(y)

∫2x³y³ dy = 2x³y⁴/4 + d(x)

Differentiating c(y) with respect to y and differentiating d(x) with respect to x, we get:

c'(y) = x³y⁴/3 + C

d'(x) = 2x³y⁴/4 + D

Since c'(y) and d'(x) are equal, we can equate them to get:

x³y⁴/3 + C = 2x³y⁴/4 + D

Simplifying, we get:

C - D = x³y⁴/12

Therefore, the potential function is:

x³y⁴/3 + C(y) = 2x³y⁴/4 + D(x)


To solve for the constant of integration, we can use the initial condition or boundary condition.

However, the equation given in the problem does not have any initial or boundary condition. Therefore, the constant of integration cannot be solved.


The general solution of the given differential equation is:

x³y⁴/3 + C(y) = 2x³y⁴/4 + D(x)

where C(y) and D(x) are arbitrary functions of y and x, respectively.


The given differential equation is exact, and its general solution is given above. However, since there is no initial or boundary condition given in the problem, the constant of integration cannot be solved.