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Jacob Jeffries

*Continued from *** 7.3 Sketching Slope Fields**.

The actual solution (which can actually be manipulated to be separable*) to the differential equation in Eq. 39 is the following:

By going to __https://www.desmos.com/calculator/fjli4efhcj__ and clicking the play button on equation 18, you can see that** this curve does indeed fit the given slope field for any constant *** C*. Clicking the play button will show different curves for different values of

The most intuitive way to think of a slope field is to picture a fluid flowing and then placing an object on the fluid that will trace out a path. This path is approximate to the solution to the curve that represents the differential equation. 😀

Fill in the table below for different values of *y’* at different coordinate points. Use a calculator to find the values to two decimal places. Create a slope field and then solve the differential equation and confirm that your slope field matches the solution to the differential equation. ✍

Varying values of C plotted over the slope field are shown here:

__https://www.desmos.com/calculator/ab0swk21ao__

*One can make this separable by doing a **substitution**:

From here, one can solve the latter differential equation (which will give a solution that is only a function of *x*) and substitute this into *u = x + y*, which will give the aforementioned solution to the original differential equation.

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