Exact Equations (Integrating Factor) - Math 210(1)

Show that the following differential equations are not exact and try to find an integrating factor to solve them. (6) To show that the given differential equation is not exact, we need to check if the partial derivatives of the equation with respect to x and y are equal. Let's calculate these partial derivatives: The differential equation is not exact since the partial derivatives are not equal. To find an integrating factor, we can use the formula: Now, let's calculate the integrating factor:
To find the integrating factor, we need to integrate this expression with respect to y: Unfortunately, this integral is quite complicated and does not have a simple closed-form solution. Therefore, we cannot find an integrating factor for this differential equation. Show that the given equations are not exact and find an integrating factor of the given form but do not solve the differential equation. (15)
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