Question

Determine whether the differential equation (y/x+5x)dx+(ln(x)−2)dy=0,x>0 is exact. If it is exact, find the solution. The differential equation exact because NOTE: Do not enter any arbitri tants. The general solutuion is =c, where c is an arbitrary constant.

Answer 1

The given differential equation (y/x+5x)dx+(ln(x)−2)dy=0,x>0 is not exact.

Step-by-step explanation:

A differential equation is exact if the total differential of a certain function can be expressed as the given equation. To determine whether the given differential equation is exact, we can check the mixed partial derivatives of the terms involving y and x. Let's denote the given equation as M(x, y)dx + N(x, y)dy = 0.

`\[M(x, y) = (y)/(x + 5x)\]`

`\[N(x, y) = \ln(x) - 2\]`

The partial derivatives are as follows:

`\[∂M/∂y = (1)/(x + 5x)\]`

`\[∂N/∂x = (1)/(x)\]`

For the equation to be exact, ∂M/∂y should be equal to ∂N/∂x. In this case, they are not equal, and thus, the differential equation is not exact.

To find an integrating factor (μ) that makes the equation exact, we can use the integrating factor formula:

`\[μ = e^{\int((∂N/∂x - ∂M/∂y)/(N))dx}\]`

After calculating μ, we multiply both sides of the differential equation by μ and check for exactness again. If it becomes exact, we proceed to find the general solution. If not, additional methods such as an integrating factor with an integrating factor can be explored. However, given the nature of this problem, further exploration may be beyond the scope of a brief explanation.