10.3. Step-by-Step Procedure

To solve using direct integration:

1. Integrate with respect to x, treating 𝑦 as constant:

$z = \int f(x, y) \, dx + \phi(y)$

where $\phi(y)$ is an arbitrary function of $y$, acting as a "constant" in integration with respect to $x$.

In the first step of solving a partial differential equation (PDE), we integrate the function $f(x, y)$ with respect to $x$. When we perform this integration, we treat $y$ as a constant to simplify the process. The integration yields a solution that incorporates an arbitrary function $\phi(y)$, which accounts for the fact that the integration is not fully complete; there may be additional functions dependent on $y$ that we do not know yet.

Think of integrating as stacking Lego blocks. You have a base (the function $f(x, y)$), and while adding blocks (integrating with respect to $x$), you keep the height (the value of $y$) constant. The extra function $\phi(y)$ is like a unique topping for each stack – it varies, but each stack still has the foundational structure.

2. If given:

$\frac{\partial z}{\partial y} = g(x,y)$

then integrate with respect to y, treating x as constant:

$z = \int g(x, y) \, dy + \psi(x)$

where $\psi(x)$ is an arbitrary function of $x$.

Similarly, if we have a PDE where the partial derivative with respect to $y$ is given, we integrate the function $g(x, y)$ with respect to $y$ while treating $x$ as a constant. This integration results in a new solution that includes another arbitrary function, $\psi(x)$, which represents the unknown components dependent on $x$.

Imagine you are filling a water bottle (the function $g(x, y)$). When you pour water in (integrate with respect to $y$), you maintain a set amount of air at the top (keeping $x$ constant). The air level varies depending on how much water you add; thus, the function $\psi(x)$ might represent this varying air space at different temperatures or pressures.

3. If both $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ are given, solve one first, then differentiate the result partially with respect to the other variable and compare with the given PDE to find the arbitrary function.

In cases where you have both partial derivatives defined, you begin by solving one PDE (either with respect to $x$ or $y$). Once you find an expression for $z$, you differentiate that expression with respect to the other variable. By doing this, you can compare the result with the original PDE. This comparison aids in identifying and determining the arbitrary function that wasn't accounted for during the first integration step.

Think of this like a detective work process. You start by finding one clue (solving for one variable) and then use that information to examine another area (differentiating with respect to another variable). By comparing notes with a partner (the original PDE), you can reveal new insights and solve for the remaining mystery pieces (the arbitrary function).

--