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### Course: Integral Calculus>Unit 2

Lesson 3: Sketching slope fields

# Worked example: slope field from equation

Given a differential equation in x and y, we can draw a segment with dy/dx as slope at any point (x,y). That's the slope field of the equation. See how we match an equation to its slope field by considering the various slopes in the diagram.

## Want to join the conversation?

• what are some applications of slope fields and how is this math used in them ?
• Slopes fields are commonly used in physics and engineering....they can also be used in biology and other life-science disciplines....For instance, they are adopted to describe predator-prey interactions! They predict how the growth rate of prey changes based on varying levels of predator population......
• So a slope is basically a cartesian plane or a graph which shows what the derivative of a graph would look like at every point on that plane? If that is true then what is the point of a slope field if you can just use the separation of variables technique to solve any differential equation?
• Only a tiny subset of differential equations can be solved by separation of variables. The slope field gives us a way to visualize and work with differential equations even if we can't solve them explicitly.
• In the case that there are multiple choices where your selected point works, shouldn't there be an easier (more efficient) way?
If dy/dx=x-y, you could just find where the slope is 0 by saying 0=x-y, which implies that y=x. Therefore, along with the line y=x, there should be horizontal lines. This gives you an infinite number of points to work with, and D is the only solution.
Was I just lucky for this problem, or would something like this work for all other problems?