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Problem 45

Slope Field In Exercises 45 and $46,$ a differential equation, a point, and a slope field are given. A slope field (or direction field) consists of line segments with slopes given by the differential equation. These line segments give a visual perspective of the slopes of the solutions of the differential equation. (a) Sketch two approximate solutions of the differential equation on the slope approximate of which passes through the indicated point. (To print field, one of which passes through the indicated point. (To print

an enlarged copy of the graph, go to Math Graphs.com.) (b) Use integration and the given point to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketch in part (a) that passes through the given point.

$\frac{d y}{d x}=x^{2}-1,(-1,3)$

Answer

$$

y=\frac{x^{3}}{3}-x+\frac{7}{3}

$$

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## Discussion

## Video Transcript

Eric. So here is the, um, float feel for why he x he won't x where minus one. And then where would you be? Wide and maybe take the integral of them. I know why eat bones. Thank you. My No, that's mean wall. Add one to the export head multiplied by the reciprocal one. That's a cue. Their power minus one x That's really one with an extra zero power. So when I add one What I haven't X Plus C and now they want us to find C and were given the 0.1. Mama three. So I love him. 34 Why I'm there. One you might and end up with three rooms. 1/3 minus one, which is negative, too. See by three plus 2/3. Add it to the other side and that gives me 7/3. We'll see. So my equation becomes. Why won't 1/3 ex huge minus X plus seven hours? And here's my soap field. You can hardly see it because it imported kind of funny, but here's my my 0.13 and here you could see how the equation matches the slope field. So this is the equation of 1/3 x cubed minus X plus 7/3 and slope field is X squared minus one

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