2.1. Linear PDEs (Lagrange's Method)

Let's start by understanding what Linear Partial Differential Equations are. They can be expressed in a standard form involving partial derivatives of a function with respect to variables.

Could you give us the standard form of a Linear PDE?

Certainly! The standard form is: $P(x,y,z)\frac{\partial z}{\partial x} + Q(x,y,z)\frac{\partial z}{\partial y} = R(x,y,z)$ where $P$, $Q$, and $R$ are functions of $x$, $y$, and $z$.

Why are they called 'linear'? What makes them different from other PDEs?

Great question! They are called linear because the dependent variable and its derivatives appear to the first power and are not multiplied together. This linearity makes them easier to solve.

Can you explain how we would go about solving these equations?

Yes! We use Lagrange's method, which involves auxiliary equations. Does anyone remember what those look like?

Isn't it $\frac{dx}{P} = \frac{dy}{Q} = \frac{dz}{R}$?

Exactly! By solving those equations, we can arrive at a general solution expressed as $\phi(u,v)=0$. This method helps us simplify complex problems. Who can summarize what we've learned so far?

We learned that Linear PDEs are in a specific form involving partial derivatives and that we can solve them using Lagrange's method!

Now, let's dive deeper into Lagrange's auxiliary equations. Who remembers how we set them up?

It's the equation we discussed: $\frac{dx}{P} = \frac{dy}{Q} = \frac{dz}{R}$.

Correct! This equation helps us express differentials in relation to one another. The next step is to solve for $x$, $y$, and $z$. What challenges do you think we might face in solving these equations?

I think it might be tricky to keep track of terms.

That’s a valid point! A solid method here is to choose appropriate functions $u$ and $v$ derived from our variables. Let’s practice a simple example. If $P = x$, $Q = y$, and $R = 1$, what do we get?

We would solve: $\frac{dx}{x} = \frac{dy}{y} = \frac{dz}{1}$.

Absolutely! Can anyone work through that and find $z$?

Integrating gives us $z = c + \ln(x) + \ln(y)$.

Well done! Recap this session for us.

We learned how to set up Lagrange's auxiliary equations and practiced finding solutions through integration!