91Ó°ÊÓ

Understanding partial derivatives is crucial in analyzing multivariable functions, which depend on multiple inputs. In the context of exact differential equations, we use partial derivatives to check for exactness.
To determine exactness, we need to identify the functions \( M(x, y) \) and \( N(x, y) \) in the differential equation.
After that, we compute the partial derivative of \( M(x, y) \) with respect to \( y \), which gives insight into how \( M \) changes with changes in \( y \), treating \( x \) as a constant.
Next, we find the partial derivative of \( N(x, y) \) with respect to \( x \), which reflects how \( N \) varies with changes in \( x \), keeping \( y \) constant.

These derivatives serve as a check for exactness. If the partial derivatives \( \frac{\partial M}{\partial y} \) and \( \frac{\partial N}{\partial x} \) are equal, the differential equation is exact.
This means there exists a potential function \( \psi(x, y) \) such that \( M(x, y) = \frac{\partial \psi}{\partial x} \) and \( N(x, y) = \frac{\partial \psi}{\partial y} \).

The potential function \( \psi(x, y) \) acts as the foundational solution for exact differential equations.
It is constructed so that its partial derivatives with respect to \( x \) and \( y \) give the original functions \( M(x, y) \) and \( N(x, y) \) of the differential equation.
To determine \( \psi(x, y) \), we integrate \( M(x, y) \) with respect to \( x \), incorporating all known terms while treating \( y \) as a constant.
This process gives a preliminary form of \( \psi \), but it might include an unknown function \( g(y) \) which accounts for terms dependent only on \( y \).
In the example, \( \int M(x, y) \, dx \) results in parts of \( \psi \), while \( g(y) \) is determined by aligning \( \frac{\partial \psi}{\partial y} \) to equal \( N(x, y) \).
The ultimate goal is to express the solution in the form \( \psi(x, y) = C \), where \( C \) is a constant. This implicitly represents the relationship between \( x \) and \( y \) that satisfies the original differential equation.

An integrating factor is a function that, when multiplied to a differential equation, makes it easier to solve by converting it into an exact equation.
Though not directly used in this exercise because the equation was already exact, it is a vital tool when the equation is not initially exact.
Finding an integrating factor might involve trial and error or more systematic approaches, such as assuming it to be a function of \( x \), \( y \), or both.
The procedure involves:

• Multiplying the entire differential equation by the integrating factor.
• Reassessing the partial derivatives \( \frac{\partial M}{\partial y} \) and \( \frac{\partial N}{\partial x} \).
• Checking if this transformation has resulted in an exact equation.

Once the equation is exact, you can find the potential function as discussed previously.

An implicit solution is a form of solution where the relationship between variables is not explicitly solved for one of the variables.
This means the solution is written as an equation involving both \( x \) and \( y \), often of the form \( \psi(x, y) = C \), where \( C \) is a constant.
In the exercise, the implicit solution is derived from the potential function \( \psi(x, y) \), and involves terms from both \( x \) and \( y \).
The resulting implicit formula can sometimes be more convenient, especially when an explicit solution would be complicated or impossible to express.

• It provides a broader characterization of the solution space.
• Implicit solutions are essential for representing complex relationships in multivariable calculus.

Adjusting \( C \) leads to different curves or surfaces in the solution space, depicting various possible solutions depending on initial or boundary conditions.