6.5. Example Problem

Solve the PDE using Charpit’s Method:

$z = px + qy + pq$

The problem requires us to solve a partial differential equation (PDE) of the form $z = px + qy + pq$. This equation represents a relationship involving the variables $x$, $y$, and their corresponding derivatives with respect to $z$, represented as $p$ and $q$.

Imagine trying to find a relationship between the amount of ingredients in a recipe (represented by $p$ and $q$) and the final dish (represented by $z$). Just like adjusting the ingredients changes the outcome of the dish, changing $px$ and $qy$ will affect the final value of $z$.

Step 1: Convert to standard form

Bring all terms to one side:

$F(x,y,z,p,q) = px + qy + pq - z = 0$

To prepare for solving the PDE using Charpit’s method, we need to rearrange the given equation into a standard form where all terms are set to zero. We illustrate this by moving $z$ to the left side of the equation, thus forming a function $F$ that equals zero.

Think of it as organizing a messy desk. We want all papers (terms) on one side (to equal zero). Once it's organized, we can identify what we have clearly and work on it more effectively.

Step 2: Compute partial derivatives

• $F_x = x + q$
• $F_y = y + p$
• $F_z = p$
• $F_p = x$
• $F_q = y$
• $F = -1$

In this step, we calculate the partial derivatives of the function $F$ with respect to each of its variables: $x$, $y$, $z$, $p$, and $q$. This will help us establish the relationships needed for the next steps in Charpit’s method.

Consider each variable as a different ingredient in a recipe. By measuring how changing the amount of each ingredient affects the final dish (which represents our function), we can understand their individual contributions.

Step 3: Write Charpit’s equations

$\frac{dx}{dt} = F_x, \; \frac{dy}{dt} = F_y, \; \frac{dz}{dt} = pF + qF + ...$

In this step, we set up Charpit's auxiliary equations, which relate the changes in $x$, $y$, $z$, $p$, and $q$ to their respective partial derivatives. These equations form a system that we will solve to find the relationship between these variables.

Imagine you are following a recipe with multiple steps. Each step depends on the previous one, just like how each equation in Charpit’s method builds upon the last. Solving each step helps you reach the final goal – the complete solution to the PDE.

Step 4: Solve the system of ODEs using any possible combination of equations.

Here, we will solve the system of ordinary differential equations (ODEs) derived from Charpit's equations. This may involve applying techniques such as substitution or integrating factor approaches. The goal is to find expressions for $p$ and $q$ in terms of $x$, $y$, and $z$.

Think of solving these ODEs as piecing together a jigsaw puzzle. Each piece (equation) must fit together correctly to reveal the overall picture (the solution to your PDE).

Step 5: Find the expressions for $p$ and $q$ in terms of $x,y,z$.

After solving the ODEs, we will express $p$ and $q$ as functions of the independent variables $x$ and $y$. This step is crucial as it links our solutions back to the original PDE.

This step is like finalizing a recipe after testing several combinations. You decide on the right proportions of ingredients (here, $p$ and $q$) for the perfect dish (the solution to the PDE).

Step 6: Integrate to obtain the complete integral (general solution) $z(x,y)$.

Finally, we integrate the expressions for $p$ and $q$ to find the complete integral, which represents the general solution of the original PDE. This step often yields a new function $z$ expressed in terms of $x$ and $y$.

Think of this as the final step in a cooking process, where all the ingredients blend together in the right proportions (through integration) to create a dish that is ready to be served (the complete solution).

This is the complete integral (general solution).

$z = ax + by + ab$

We have now derived the complete integral of the PDE, which is the general solution expressed as $z = ax + by + ab$, where $a$ and $b$ are constants. This form signifies that the solution can take many shapes depending on the values of these constants.

Like a customizable recipe where the base dish is the same (our solution), but the flavor changes with different spices (constants $a$ and $b$). Everyone can create their unique version based on their taste!

--