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A direction field is drawn for dy/dx = x² - y…

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A direction field is drawn for $\dfrac{dy}{dx} = x^2 - y$ . Find the slope of the segment at $(3, 2)$ and at $(1, 3)$ . Then find the value of $y$ at $x = 1$ where the segment is flat.

Need a reminder? The key ideaShow

The equation $\dfrac{dy}{dx} = f(x, y)$ gives the slope at every point. A solution curve follows these slopes, and two curves never cross. A step along the field is

\displaystyle y_{\text{new}} = y_{\text{old}} + h\,f(x_{\text{old}}, y_{\text{old}}).

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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A direction field is drawn for dy/dx = x² - y…

The worked solution

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