- Difficulty
- Challenging
- Suggested time
- 5 min
- Marks
- 3
Slope fields and Euler’s method practice question 10
Exactly one of these straight lines is itself a solution curve of . Which one?
A straight line has the same slope everywhere. For it to be a solution curve, what must the field’s slope be at every point along it?
A line is a solution when its own slope equals the field’s slope at every , not just at one point.
Write the condition as for all : the coefficient of and the constant term must match separately.
Theory UBC Math 100 §12.2Key results
Worked solution
Strategy
Substitute each line into both sides of the equation. A curve is a solution only if its slope agrees with the field at every point along it.
Working
The two sides agree for every t.
The field is flat along this line but the line climbs, so solutions cross it.
These agree only at t = 0; one matching point is not enough.
They disagree everywhere.
Every other solution has this form (substitute to check), so all solutions approach the line y = 2t − 2.
Answer
The answer is y equals 2 t minus 2.
Check
A line y = mt + c needs m = (2 − m)t − c for every t, which forces m = 2 and c = −2.
Where this goes wrong
Choosing y = 2t, the line where the slopes are zero. Zero slopes there mean solutions are momentarily flat as they cross it; the line itself rises, so it cannot be a solution.
Leans on
The slope of a straight line, from Functions, graphs and inverses
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