The idea
A differential equation specifies a relationship involving an unknown function and its rates of change. An initial value selects a particular solution.
Worked foundation example
A solution satisfies dy/dx=x+y. What is its slope at (2,3)? A slope field evaluates the differential equation at the specified point. Substitute x=2, y=3. The slope is 5.
The common trap
A numerical approximation is not an exact symbolic solution. In a slope field, a slope depends on the coordinates, not on the visual length of the segment.
Connect the representations
Check a proposed solution by substitution into both the differential equation and the initial condition. Checking only one is insufficient.
Explain before checking
Write the defining equation, state its assumptions, and attach units before calculating. After checking the numerical answer, compare every step with the worked reasoning. These short tasks build selected foundations; a complete AP free-response solution also needs written justification, correct notation and attention to the rubric.
Compare a general solution with an initial-value solution
For y′=2xy and y>0, separation gives dy/y=2x dx and hence ln y=x²+C. Exponentiating gives y=Ke^(x²), where K is positive. If y(0)=3, then K=3. Verify both parts: differentiating 3e^(x²) gives 2x times the function, and its value at zero is 3. Satisfying the differential equation alone does not guarantee the initial condition.
Transfer and explain
Transfer: for y′=4x and y(0)=3, the solution is 2x²+3. This equation integrates directly because its right side has no y factor. Euler approximation is BC-only and is not part of this AB practice pack.
A solution satisfies dy/dx=x+y. What is its slope at (2,3)?
- A slope field evaluates the differential equation at the specified point.
- Substitute x=2, y=3.
- The slope is 5.
Compare the other answer choices
Ignore y in the slope law.
Ignore x in the slope law.
Multiply x and y rather than add.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Calculus AB courseCalculus Volume 1: Fundamental Theorem of Calculus