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Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point.$$y^{\prime}=x+y^{2}, \quad(0,0)$$
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Calculus 2 / BC
Chapter 9
Differential Equations
Section 2
Direction Fields and Euler's Method
Campbell University
Baylor University
Idaho State University
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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Sketch the direction field…
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Sketch a direction field f…
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Were given the differential equation in the fort an when we asked to find the silk field dissipation and to draw the graph associate contains this point? The traction equation is y prime equals x plus y there, and the point of an is origin 006 of this equation. To season of numbers that y print is equal to c, there follows that c gonna be equal to x, plus y squared x, is equal to negative y square plus c. You can think of these as horizontal holabalos that lie on the x axis. Basing for the negative in time and which have an x intercept a 0, can we have the appoints on this careless by bronxies on to emphasize the negative side of the x axis? Is that's what most of the parable ogil beso in particular? If c is equal to 0, then we have x equal to negative y, so we have gravelotte point at 00 and also as a point that negative 11 and negative 1 negative 1 point now. On this part, the points have slope yeea to c to be so 1. The more have x equals negative y 3 plus 1, and we have a .10 on this gravity, which is a slope 1. We also have the .01 and 0 negative 1. No long is parabolas, and this caramels just outside my first pathen, see that the c gets larger and larger opeongo deeper and deeper, and so you get these shoals of parabolis, which we see brace a persian. The slump is still always are vertical, or that for the present now i see is some in between 0 and 1 we ginawater, but it's look to be between 0 and 1, so in between the slopes of the 2 blades, let's in it, if c is going To be less than 0 and the slope is going to be less than 0, the prater will have its apex at point c, less than 0, so the game will have. A series of brown was a inch which the slopes are gradually decreasing. So you see that as you get further and further away towards the negative, the deity on the x axis, the slope field again sequent later nearly weak on that axis. So this is a rough representation at slopes liteiso, this wiles, a bunch of nested parabolis, which the light down on the slopes gets more and more negative as we go towards negative at on the x axis .00, the organs contare- and you see that as we approach Positive infinity or canine the function increases positive exponential and, as we approach negative infinity there so along the x axis, we have x is equal to 0 y prime is equal to y. Squared y is equal to 0. We have y prime is equal to x or that in all that y would be equal to 1 half of x, squared so more nearly steeper and steeper the silk etsessentially. What we have is presentation it as x, approaches, negative infinity on this function said to be y, is equal to 0 in a thin integrality, 1 half x square plus c percent cimon this. Now we go .0 solution. We a 60 y set 1 half x square, and so we have that the slopes of these points by prime is going to be xiraena essentially sent. The function is being pushed upwards, and so we get a point where he is 0 slope and then again to be pushed downwards again. So this is 1 direction. They can go atter curve. That'S any possible is, are going up, go down and follow this other parable, which has soporoand on the other sidethis is our connected. That'S actually, that's a.
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