- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Slope fields and Euler’s method practice question 7
The table gives the slope of the segment drawn at each point of a slope field. Which differential equation was the field drawn for?
Read along each row: the slope does not change as changes. What does that say about whether appears in the equation?
Then test the remaining candidates at and , where the slopes are 1 and .
Theory UBC Math 100 §12.2Key results
Worked solution
Strategy
Read the pattern before computing. Slopes that are the same all along every row mean the right-hand side depends on y alone; then check the candidates row by row.
Working
Both of these change along a row, so neither can produce a table whose rows are constant.
The signs are the reverse of the table.
Matches every row. The rows y = 0 and y = 2 are flat, so those horizontal lines are themselves solutions.
Answer
The answer is dy by dt equals y times the quantity 2 minus y.
Check
Flat rows at y = 0 and y = 2 call for factors y and y − 2, and the slope +1 at y = 1 picks 2 − y over y − 2.
Where this goes wrong
Checking only the t = 0 column, which (y − t)(2 − y) matches as well. Slopes that stay the same along each row say that t does not appear at all.
Leans on
The derivative as the slope of a tangent line, from The definition of the derivative
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