To form a partial differential equation (PDE) by eliminating the arbitrary functions $z = f(2x + 3y) + g(2x + y)$, we need to find the derivatives of $z$ with respect to $x$ and $y$, and then eliminate the arbitrary functions $f$ and $g$.

Step 1: Differentiate with respect to $x$

First, differentiate $z = f(2x + 3y) + g(2x + y)$ with respect to $x$:

$\frac{\partial z}{\partial x} = f'(2x + 3y) \cdot \frac{\partial}{\partial x}(2x + 3y) + g'(2x + y) \cdot \frac{\partial}{\partial x}(2x + y)$

Since $\frac{\partial}{\partial x}(2x + 3y) = 2$ and $\frac{\partial}{\partial x}(2x + y) = 2$, we get:

$\frac{\partial z}{\partial x} = 2 f'(2x + 3y) + 2 g'(2x + y)$

Step 2: Differentiate with respect to $y$

Now, differentiate $z = f(2x + 3y) + g(2x + y)$ with respect to $y$:

$\frac{\partial z}{\partial y} = f'(2x + 3y) \cdot \frac{\partial}{\partial y}(2x + 3y) + g'(2x + y) \cdot \frac{\partial}{\partial y}(2x + y)$

Since $\frac{\partial}{\partial y}(2x + 3y) = 3$ and $\frac{\partial}{\partial y}(2x + y) = 1$, we get:

$\frac{\partial z}{\partial y} = 3 f'(2x + 3y) + g'(2x + y)$

Step 3: Eliminate the arbitrary functions $f$ and $g$

From the above two derivatives, we have: $\frac{\partial z}{\partial x} = 2 f'(2x + 3y) + 2 g'(2x + y)$ $\frac{\partial z}{\partial y} = 3 f'(2x + 3y) + g'(2x + y)$

Now, to eliminate $f'(2x + 3y)$ and $g'(2x + y)$, we can solve these equations simultaneously.

Step 4: Solve the system

Multiply the first equation by 3, and the second equation by 2:

$3 \frac{\partial z}{\partial x} = 6 f'(2x + 3y) + 6 g'(2x + y)$ $2 \frac{\partial z}{\partial y} = 6 f'(2x + 3y) + 2 g'(2x + y)$

Subtract the second equation from the first:

$3 \frac{\partial z}{\partial x} - 2 \frac{\partial z}{\partial y} = (6 f'(2x + 3y) + 6 g'(2x + y)) - (6 f'(2x + 3y) + 2 g'(2x + y))$

This simplifies to:

$3 \frac{\partial z}{\partial x} - 2 \frac{\partial z}{\partial y} = 4 g'(2x + y)$

Now, we express $g'(2x + y)$ in terms of the derivatives of $z$:

$g'(2x + y) = \frac{1}{4} \left( 3 \frac{\partial z}{\partial x} - 2 \frac{\partial z}{\partial y} \right)$

This is a partial differential equation that eliminates the arbitrary function $g$, and we can then proceed to a similar elimination for $f$, if desired.

Final PDE Form:

The equation formed is:

$3 \frac{\partial z}{\partial x} - 2 \frac{\partial z}{\partial y} = 4 g'(2x + y)$

This is the general form of the PDE, eliminating the arbitrary function $f$ and $g$ by relating their derivatives.

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5 Related Questions:

1. How can we eliminate $f$ and $g$ from the equation in terms of higher-order derivatives?
2. Can we derive a specific solution to this PDE given boundary conditions?
3. What is the method to convert this PDE to a canonical form?
4. How do we solve PDEs involving arbitrary functions like $f$ and $g$?
5. What is the role of the method of characteristics in solving such PDEs?

Tip: When eliminating arbitrary functions from PDEs, look for systematic ways to use the total derivatives and exploit symmetry between the variables.