The equation dy/dx = x - y has the solution y = x - 1 + Ke⁻ˣ…

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The equation $\dfrac{dy}{dx} = x - y$ has the solution $y = x - 1 + Ke^{-x}$ . The curve passes through $(0, 1)$ . Find $K$ and then the heights $y(1)$ and $y(2)$ to four decimal places. The curve approaches the line $y = x - 1$ . Find how far above the line the curve is at $x = 4$ , to four decimal places.

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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The equation dy/dx = x - y has the solution y = x - 1 + Ke⁻ˣ…

The worked solution

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