In a mathematical programming model, if the objective functions and constraints are represented entirely by linear equations, then the model is regarded as a linear model.

The whole point in learning differential equations is that eventually we want to model real physical systems.

Whenever we solve these differential equations analytically, we're really just trying to get a pure model of something.

That's true with regular equations or differential equations.

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both equations.

Exact equations.

Separable equations.

And you're like, Sal, you just said that differential equations, we're learning to model things, why is this type of a function useful?

Partial Differential Equations Index at EqWorld The World of Mathematical Equations.

Partial Differential Equations Methods at EqWorld The World of Mathematical Equations.

have some number of equations, say n equations and n unknowns.

of linear equations.

Math, differential equations.

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linear equations right here.

So physicists write equations.

Systems of linear equations.

Three equations, three unknowns.

So the differential equations.

External links Partial Differential Equations Exact Solutions at EqWorld The World of Mathematical Equations.

I was able to reduce this system of equations to this system of equations.

Partial Differential Equations in Cleve Moler Numerical Computing with MATLAB Example problems with solutions at exampleproblems.com Partial Differential Equations at mathworld.wolfram.com Partial Differential Equations at nag.com Dispersive PDE Wiki NEQwiki, the nonlinear equations encyclopedia

And this first class of differential equations I'll introduce you to, they're called separable equations.

And when you study exact equations in differential equations, you'll see this a lot more.

Mathematically speaking, the simplification reduces the state space of the system to a finite dimension, and the partial differential equations (PDEs) of the continuous (infinite dimensional) time and space model of the physical system into ordinary differential equations (ODEs) with a finite number of parameters.

Two equations and two unknowns.

You encode it into equations.

linear equations, x and y.

least the solvable differential equations.

Two equations with two unknowns.

Three unknowns with three equations.

Let's add the two equations.

So what were linear equations?

That's why they're called 'equations.'

You encode it into equations.

I have separated the equations.

So both of these parametric equations, both sets of parametric equations have the exact same path.

The equations in contexts outside of general relativity are still referred to as the Einstein field equations.

But hopefully you might see the parallels if you've already run into exact equations in differential equations.

linear differential equations with constant coefficients.

Two linear equations of to unknowns.

Let's just add these two equations.

Let's do some more nonhomogeneous equations.

And all the equations break down.