91Ó°ÊÓ

A partial differential equation (PDE) involves functions and their partial derivatives. In a first-order PDE, the highest derivative is of the first order. In this exercise, we have the PDE \( u_{y} + 2y u = 0 \), where we are only differentiating with respect to the variable \( y \). This indicates it's a first-order linear PDE.To identify a PDE of this kind, note:

• It involves only first derivatives.
• The dependent variable is linear, which means it appears to the first power and there's no multiplication between the derivatives.
• Coefficients can be functions of both independent and dependent variables.

In this PDE, the variable \( x \) acts as a parameter, meaning it does not influence the derivative directly. Such a PDE acts very much like an ordinary differential equation (ODE) with respect to the independent variable \( y \), allowing us to use techniques typically reserved for ODEs.

When faced with a first-order linear differential equation like the one in our exercise, the integrating factor method is a useful technique. This method simplifies the process of solving by turning the left-hand side of the equation into an exact derivative, making integration straightforward.Here's how the method works:

• First, identify the function to be used as the integrating factor. For an equation of the form \( \frac{du}{dy} + P(y)u = 0 \), compute \( \mu(y) = e^{\int P(y) \, dy} \).
• In this equation, \( P(y) = 2y \), leading to \( \mu(y) = e^{y^2} \).
• Multiplying the entire differential equation by \( \mu(y) \) helps combine the terms into the derivative of a single expression.

This integrating factor effectively transforms the problem into one that is much easier to integrate directly, resulting in an equation that expresses a relationship between \( u \) and the function of \( y \) we've created.

An ordinary differential equation (ODE) is a relationship between an unknown function and its derivatives. In the context of the exercise, the PDE resembles an ODE because it only involves derivatives with respect to one variable, \( y \). Consequently, the solution strategy borrows heavily from ODE techniques.The transformed PDE, \( \frac{du}{dy} + 2yu = 0 \), is a linear first-order ODE with respect to \( y \). Understanding the characteristics of ODEs helps in solving such equations:

• The solution involves integration with respect to the variable \( y \).
• It allows finding a function that satisfies the equation for given initial or boundary conditions.
• As per the integrating factor method, post-integration, we solve for the function \( u(x,y) \) in terms of \( y \) and parameterized by \( x \).

By using ODE solution techniques, we arrive at the general solution where \( u(x,y) = \frac{C(x)}{e^{y^2}} \). The arbitrary function \( C(x) \) arises because of the parameter nature of \( x \) in the context, akin to the arbitrary constant in standard ODE solutions.