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Worked example: range of solution curve from slope field

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Given the slope field of a differential equation, we can sketch various solutions to the equation. In this example, we analyze the range of a specific solution.

Video transcript

- [Instructor] If the initial condition is zero comma six, what is the range of the solution curve y is equal to f of x for x is greater than or equal to zero? So we have a slope field here, for a differential equation. And we're saying, okay, if we have a solution where the initial condition is zero comma six, so zero comma six is part of that solution. So let's see, zero comma six, so this is part of the solution. And we want to know the range of the solution curve. So solution curve, you can eyeball a little bit by looking at the slope field. So as x, remember, x is gonna be greater than or equal to zero, so it's going to include this point right over here. And as x increases, you can tell from the slope, okay, y is gonna decrease, but it's gonna keep decreasing at a slower and slower rate. And it looks like it's asymptoting towards the line y is equal to four. So it's gonna get really, as x gets larger and larger, larger, it's gonna get infinitely close to y is equal to four, but it's not quite gonna get there. So the range, the y-values that this is going to take on, y is going to be greater than four. It's never gonna be equal to four. So I'll do, it's going to be greater than four. That's gonna be the bottom end of my range. And at the top end of my range, I will be equal to six. Six is the largest value that I am going to take on. Another way I could have written this is four is less than y is less than or equal to six. Either way, this is a way of describing the range. The y-values that the solution will take on for x being greater than or equal to zero. If they said for all x's, well, then you might have been able to go back this way and keep going. But they're saying the range of the solution curve for x is greater than or equal to zero. So we won't consider those values of x less than zero. So there you go, the curve would look something like that. And you can see, the highest value it takes on is six, and it actually does take on that value 'cause we're including x equaling zero. And then it keeps going down, approaching four, getting very, very close to four, but never quite equaling four.