Practice a College Board-style free response question on slope fields and separation of variables. Write your response, then reveal the model answer to see exactly what earns each point.
Free Response Question · Unit 7 · Differential Equations
Consider the differential equation dy/dx = 2xy, with the particular solution y = f(x) that satisfies the initial condition f(0) = 3.
A
Find the value of dy/dx at the point (1, 2), and describe what the slope field looks like along the y-axis (x = 0).
✓ Model answer
At (1, 2): dy/dx = 2(1)(2) = 4. Along the y-axis, x = 0, so dy/dx = 2(0)(y) = 0 for every y — the slope field segments are horizontal all along x = 0.
Why it scores: Correctly evaluates dy/dx = 4 at the point, and recognizes that x = 0 forces zero slope (horizontal segments) regardless of y.
B
Use separation of variables to find the general solution of dy/dx = 2xy.
✓ Model answer
Separate: (1/y) dy = 2x dx. Integrate both sides: ln|y| = x² + C. Exponentiate: |y| = e^(x² + C) = eᶜ·e^(x²), so y = Ae^(x²), where A = ±eᶜ is an arbitrary constant.
Why it scores: Separates correctly, integrates to ln|y| = x² + C, and solves for y = Ae^(x²). Losing the constant or mis-integrating loses points.
C
Find the particular solution y = f(x) satisfying f(0) = 3.
✓ Model answer
Using y = Ae^(x²) with f(0) = 3: 3 = A·e⁰ = A, so A = 3. The particular solution is y = 3e^(x²).
Why it scores: Substitutes the initial condition to find A = 3 and states y = 3e^(x²). Forgetting to apply the initial condition, or leaving the answer with an unknown constant, loses the point.
How to score points on AP Calculus FRQs
Evaluate dy/dx directly at a point for slope-field questions.
Separate variables cleanly — y-terms with dy, x-terms with dx — then integrate.
Always include + C and apply the initial condition to find it.
Solve explicitly for y when the question asks for y = f(x).
Answer the verb. "Find" wants the solution; "describe" wants the behavior.