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A direction field for the differential equation $ y^1 = x \sin y $ is shown.(a) Sketch the graphs of the solutions that satisfy the given initial conditions.(i) $y(0)=1$(ii) $y(0)=0.2$(iii) $y(0)=2$(iv) $y(1)=3$(b) Find all the equilibrium solutions.

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Calculus 2 / BC

Chapter 9

Differential Equations

Section 2

Direction Fields and Euler's Method

Missouri State University

Baylor University

University of Nottingham

Lectures

13:37

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

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A direction field for the …

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we're giving a differential Europe sketch were asked my solutions given initial value Find all livers, The equation getting a white one high want and associates Slope field given right heart A You're asked to fine rafts of solutions. Given initial value condition part I Y zero equals one. No, that's graphic Black graph Will have a 0.1 following. No. Yes, you get. This is a where everyone inside but right is more deeply fact that hands you attitudes when it white zero and you see that this raft does not have any value. First, quantum quality work for next part 25 value. Why zero? Do you now grab this in red. We know you have points at zero to you were working and again nearly here he gets another graveled on like ocean. Do you think, Charles? Asking like zero. And it looks graph from part I or one over you say part three were asked to find a solution. Initial value wise. Zero people's a graph this green. Do you have a 0.2 and you see that all of this is simply or dump a warrant to in part for there has to find a solution. Initial value. One of one graphics again. So we have that 13 Why? And falling slope You have sort of fourth graf It has asked takes this time at wife to four. Something like a translation of the graph. Parts 12 but burning off the Y axis and also in part B. You're fine. Oh, this looking at you for his uncle Acid codes at zero y equals two and white, four like desk. All equilibrium solutions are stable. Horizontal lines which have form y equals two k k. It's like, well, something in number of this, we can use the equation. So supposing that why we'll see isn't everything where some see in real number then we have that. Why climb? Because zero do you have ocean Indian? 1/2 high winds. Ford, however, to seen is one of the equal to y zero. You know that the Indians is zero whenever sign is zero. This is whenever Russian inside Hi over to sign is zero or anything. You're more full time insulting, perceived get season two to So we have shown that they have any living solution. That solution must have the form y two or some? Oh, sure. This isn't even report from solution. What is it for you? So I assume that why is, um we're taking this as a possible solution? Why? You know, uh, I have to y and is equal of in pie. Yeah. Zero in there for a while. So we see. Yeah. Uh, solution yplan people's, uh I have to. Why? Set of Old Ocean? Why? Home isn't.

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