"The linearity of a differential equation depends on linearity in the dependent variable and its derivatives."
"You've got two avenues to go around. One is when you know the differential equation and parameters involved, you use Simulink. Or, you can string up SimScape blocks with physical connections and build your model out that way."
"Bernoulli equations make the differential equation linear, making it easier to solve."
"Any second-order differential equation can be written as two coupled first-order ODEs."
"Linearizing or linearization... a big step transitioning from... simple linear ordinary differential equations to starting to build the framework... for solving... very non-linear systems."
"Solving a differential equation means: from an equation that just contains the increments going to an entire function."
"For all subsequent cars... they are described by this differential delay equation."
"My entire simulation was just doing Euler's method, which is a numerical simulation on solving first-order differential equations."
"The golden functions of differential equations are the ones where you know their Laplace transform and you can go back and forth easily."
"A partial differential equation or a PDE is basically an equation involving one or more derivatives of an unknown function of two or more independent variables."
"It lets you differentiate through systems of differential equations in the time that costs basically the cost of evaluating one additional time."
"Ordinary differential equations are much easier to solve."
"Probably the most important thing about the Laplace transform is that you can use it to solve differential equations."
"That is just the most basic introduction to differential equations."
"If you do these things together, then you'll be able to solve your differential equation quite efficiently."
"Differential equations are a discipline where performance really matters."
"That's the most basic introduction to differential equations."
"The trajectories which solve these equations are known as geodesics."
"So the last few lectures I have shown you how to solve systems of differential equations with forcing."
"Differential equations are really everywhere, every branch of engineering, every branch of physics, mathematics, chemistry even."
"The work of Cartwright and Littlewood... changed the way we think about differential equations forever."
"The solution is formed of the inhomogeneous part and then the homogeneous solution or complementary function."
"Integrating factors are used to solve first order linear differential equations."
"We can also start to think about how to solve partial differential equations on curved surfaces."
"This right here is the standard form of a second order differential equation."
"Complex numbers are extremely important in differential equations."
"I solved a differential equation... without calculus; all I did was polynomial math."
"This is a tutorial on how to solve differential equations in Python."
"So I'm pretty excited we are starting to get into my favorite material in differential equations."
"On its own, the left-hand side of this differential equation equaling 0 is just the simple harmonic oscillator with a sinusoidal solution."
"The properties of the Fourier transform provide us with a very useful and important mechanism for solving linear constant coefficient differential equations."
"This is our final particular solution for this differential equation."
"We're going to solve another differential equation, ratcheting up the complexity a little bit as we go."
"Let's not get too far ahead of ourselves; let's look at the differential equation."
"The initial condition is x at 0 is equal to 0."
"There's only a first derivative anywhere in this equation, that means there's only one initial condition required to solve it."
"If you remember, the Laplace transform of the first derivative is s times the Laplace minus the initial condition."
"When you solve a differential equation, what you're trying to do is figure out what X of T is."
"So I fell in love with differential equations, it opened up a whole world of possibilities, of things you could model, of things you can understand, of things you can build with math with equations."
"It's so powerful and versatile, you're going to use e to the X as the basis for the solution to all linear differential equations."
"Solving new differential equations becomes very, very easy."
"A standard form of a linear DE is the following: dy/dx plus P of x y equals f of x."
"When you multiply mu of x, this whole thing will always become DDX of mu of x times y."
"After you do this, the whole left-hand side of your DE magically will become this."
"The finite difference method is a numerical method used to solve ordinary differential equations and partial differential equations."
"A linear differential equation is simply one which has this very special form."
"We've been talking about linear systems of differential equations like $$ \dot{x} = Ax $$, where A is a matrix and X is a vector."
"Remember, our characteristic polynomial for y double prime plus three y prime plus two y equal to zero."
"The eigenvalues of this A matrix determine everything about the solution to this differential equation."
"So, if you know what type of DE it is, and you can take the right approach, that's huge."
"This is your very standard differential equation here which describes this block."
"Roncada methods are a family of approaches to numerically solve the ordinary differential equations."
"Differential equations become algebraic equations in the Laplace domain."
"We're almost done exploring what differential equations are and what the solutions look like."
"A lot of differential equations use approximations because there aren't any good techniques to solve some of these things we're going to come up against."
"What a reducible second-order differential equation means is that we can change the second-order into a first-order."
"Variation of parameters... is a surefire method for finding a particular solution of a linear differential equation."
"These matrices T and D are not magic, they are very simple and intuitive matrices that are useful for solving systems of differential equations."
"Upon eliminating the arbitrary constants, we will be able to determine the differential equation of that certain equation."
"The differential equation for this function is equal only to the third derivative with respect to X is equal to zero."
"They're connected by the Cauchy-Riemann equations."
"Maximum principles can be used all over the place because a lot of things in mathematics are described using partial differential equations."
"This works if we have a second order differential equation where we are missing a variable completely."
"We're about to get there again, videos before that, but right now it's kind of this thought like, are there techniques to deal with higher order differential equations? Yes, of course, there are."
"In the realm of differential equations, the conditions dictate the solution, not the equations themselves."
"Finding the differential equation of a family of curves is like uncovering the DNA of geometry."
"Find the differential equation of the family of lines passing through the origin."
"An initial value problem is a differential equation that comes with some initial conditions."
"This is the particular solution of the given differential equation."
"This is now our general solution."
"In the next video, I will be discussing a new topic which is about an introduction to the modeling process using differential equations."
"I will teach you how to formulate differential equations from real-life situations."
"A differential equation is said to be homogeneous if every term has the same degree."
"If every term has reached a certain exponent n, then definitely we have a homogeneous differential equation."
"We're going to start learning how to actually solve some differential equations from scratch without being given the solution."
"In setting up a differential equation, we're going to be defining a new type of relationship that we haven't looked at before."
"Differential equations most of the time is concerned with approximating good enough so that it's useful."
"I hope this is making sense, we should hit in the next video to get onto another technique of solving differential equations which is pretty cool."
"The goal of math is to get it to apply to things; otherwise, it's fun, and studying pure math can be fun, but we really only confer usefulness in differential equations."
"A differential equation is an equation that has both a function and one or more of its derivatives in it."
"The formation of differential equations from known solutions makes the topic more propulsive."
"By eliminating the arbitrary constant, we will be left with the desired differential equation."
"This is a linear differential equation and since we have f(x) not equal to zero, this is non-homogeneous."
"A differential equation is an equation that contains an unknown function and some of its derivatives."
"The order of a differential equation is the order of the highest derivative that occurs in the equation."
"A function f is called a solution of a differential equation if the equation is satisfied when y equals f of X and its derivatives are substituted into the equation."
"Y equals c1 e to the negative 3x plus c2 e to the 6 X is a general solution of y double prime minus 3 y prime minus 18y equals 0."
"The solution here to this differential equation would be a set of differentiable functions X and Y, and they're both functions of T."
"The Euler-Lagrange equation is the strong form for a 1D problem."
"We can use this to solve differential equations."
"We will move on to the solution of ordinary differential equations."
"Use cases for this functionality include curve fitting, parameter estimation in partial differential equations, and other inverse modeling problems."
"If you've understood how to get these graphs from your differential equation, you're doing great."
"This is the first video in a series devoted to a first course in differential equations, namely ordinary differential equations."
"We have monotonic convergence to the solution of the problem-governing differential equations."
"The idea behind finite elements is it's a mathematical technique used to convert a partial differential equation into a system of ordinary differential equations."