Determine whether the given differential equation is exact. If it is exact, solve it. (5x + 4y) dx + (4x - 8y³) dy = 0.

Solution:

A differential equation is an equation that contains at least one derivative of an unknown function, either an ordinary derivative or a partial derivative.

Suppose the rate of change of a function y with respect to x is inversely proportional to y, we express it as dy/dx = k/y.

Given, the differential equation is (5x + 4y) dx + (4x - 8y³) dy = 0.

We have to determine the solution.

By multiplicative and distributive property,

5xdx + 4ydx + 4xdy - 8y³dy = 0

= d(5x²/2) + d(4xy) - d(2y⁴)

= 5x²/2 + 4xy - 2y⁴ = C

On integrating,

\(\int M\, dx=\int (5x+4y)dx=\frac{5x^{2}}{2}+4xy+f(y)\)

On differentiating,

\(\frac{d}{dy}\int M\, dx=4x+f'(y)=4x-8y^{3}\)

f’(y) = -8y³

Now, f(y) = -2y⁴ + C

Therefore, the solution is 5x²/2 + 4xy - 2y⁴ = C.