Question

Verify that the diferential equation

(5x + 4y)dx + (4x − 8y^3)dy = 0

is exact and find its solution curves.

Answer 1

The answer is "
`C= (5)/(2x^2)+ 4xy - 2y^4.`"

Explanation:

It given that,
`\frac {\partial(5x + 4y)}{\partial y} = \frac{\partial(4x- 8y^3)} {\partial x} \\\\` it is the differential equation.

Solve the above equation.

`\frac {\partial M (x, y)}{\partial y} = (5x+ 4y)`

So,
`M (x, y) = (5)/(2x^2)+4xy +\varphi (y).`

`(\partial M (x, y))/( \partial y )= 4x- 8y^3 \ and \ \ 4x + \varphi ' (y) = 4x- 8y^3 \\\\\ so, \ \\varphi ' (y) = -8y^3`

or
`\varphi (y) = -2y^4+c`

To find the general solution

`M(x, y )= c \\C= (5)/(2x^2)+ 4xy - 2y^4.`