A drawing compass is placed with its needle at the point (a, 0) on the x-axis, and the pencil is…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A drawing compass is placed with its needle at the point on the -axis, and the pencil is set so that the circle passes through the origin. As the needle moves along the axis, the compass draws a family of circles , where is a different constant for each circle. The designer wants one differential equation that every circle of the family satisfies, with no in it. She also wants the slope of the circle with at the point . State the order and the degree of the equation, and find the slope at to four decimal places.
Need a reminder? The key ideaShow
To form the equation of a family with constants, differentiate times and eliminate the constants. The result has order . Read its degree after clearing roots.
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.
Given. The family , with one constant .
Plan. One constant means order . Differentiate once and eliminate .
Step 1. Differentiate. , so .
Step 2. Eliminate . Put it in the family: . Then , which gives
Step 3. Order and degree. The highest derivative is , and its power is . So the order is and the degree is .
Step 4. The slope. The point lies on the circle with , since . The slope is .
Answer: order , degree and slope .
Check. Differentiating the circle directly gives , the same value.
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