Welcome to my course on differential equations. In this video, I want to tell you some of the terminology associated with differential equations. Here I have written three types of differential equations on the board. The first one is the equation for the RLC circuit in electrical engineering. Then the second one is the equation of motion for a pendulum, a mass that is oscillating back and forth. The third one is the diffusion equation which governs the motion say of pollution dispersing in the air. These are all what are called second-order differential equations, because the order of a differential equation is determined by the order of the highest derivative. So, the first equation has a second derivative of q with respect to time. The second equation, the highest order derivative is the second derivative of Theta with respect to time. The last equation we have a partial squared u with respect to x squared et cetera which is also second order derivatives. You also notice that the first two equations just have an ordinary derivative in them because there is only one independent variable which here is t. So, q is just a function of t, q is the charge on the capacitor in the circuit. Theta is just a function of t, Theta is the angle that the pendulum makes with the vertical. But in the third equation, u which is the concentration of the pollution in the air, is a function both of position x, y and z and also t. So, the first two equations are what we call ordinary differential equations or ODEs. The third differential equation is what we call partial differential equations or PDEs. In this course, we'll spend most of our time studying ODEs but in the end we'll tackle a PDE which will be a simplified version of this diffusion equation. There's another dichotomy of these equations, that's the linear or non-linear type of equation. In this course, we'll be focusing on linear equations because those are the equations that have analytical solutions. That's what we'll be doing in this course, is finding analytical solutions. But the non-linear equations are also very important for modeling problems in engineering and physics and other fields. So how about these three equations? The first equation is a linear equation. By linear, we mean linear in q so, there's wherever q enters, it enters by itself. So, there is a d squared q, dt squared, a dq, dt and a q. There is never a q squared, there is never a function of q that has terms in the Taylor series that are above q. In that case, then it's called linear. The coefficients can be functions of time in a linear equation, but you must only have q by itself. The third equation is also a linear equation. Here it's linear in u, so there is no u squared term in this equation either. However, the second equation is a non-linear equation. There is a d squared Theta, dt squared. There's a d Theta, dt. Those two terms are linear but then there's this term here, which is sine Theta. So sine Theta has a Taylor series of Theta minus Theta cubed over three factorial. So, it has terms of higher powers of Theta in it and this is called a non-linear term in this equation. So, this term is considered a non-linear equation. Okay, so in this course, we'll do first-order differential equations, second-order differential equations. Mainly linear equations, we'll focus on linear equations. We'll do mainly ordinary differential equations except for the last topic of this course, where we'll tackle the solution of a partial differential equation. So, I think in this first lecture, I should at least solve a differential equation. There is a very simple differential equation you can solve already just using calculus. That's the equation for a mass falling under gravity without any air resistance, that equation would take the form of d squared x, dt squared equals minus g. This equation says that the acceleration of that mass is a constant and in the downward direction, g is the well-known 9.8 meters per second squared. So, this type of equation because there is no other x dependence besides the derivative term and there is no t dependence in this equation, means you can solve this equation just by integrating. So, if you integrate this equation twice, you would end up with a solution that I can write down may be familiar to many of you. X of t is equal to the initial position of the mass plus the initial velocity of the mass times t minus 1.5 gt squared. Okay, that's just by integrating this equation twice. You can see that the first derivative of putting t equals 0 and this equation gives you x naught. That's called an initial condition. So, the initial conditions here, we'll see that a second-order differential equation needs two initial conditions. The initial conditions here is simply that x0 is equal to x naught. Then if you take the first derivative of this equation, you have dx, dt at 0 is equal to u naught, that would be the initial velocity of this mass. Then the minus 1.5 gt squared is enough to satisfy that the second derivative of x is equal to minus g Okay. Let me recap, I'm introducing some terminology associated with differential equations. The order of the equation, is the order of the highest derivative in the equation. You have ordinary differential equations or ODEs and partial differential equations or PDEs. You have linear and non-linear differential equations. In this course we'll focus on linear equations because they are the ones that you can solve using analytical methods. Finally, I solve a differential equation which everyone should be able to solve coming from a calculus course and that's just the saying that the acceleration is a constant. I'm Jeff Chasnov thanks for watching and I'll see you in the next video
