The differential equation is to be analyzed graphically.
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
The differential equation dxdy=x2−y is to be analyzed graphically.
(a) Show that all points satisfying y=x2 have slope 0. Explain why this set of points is useful when constructing a slope field.
(b) On a grid containing the rectangle −3≤x≤3 and −2≤y≤7, sketch a slope field that clearly distinguishes regions of positive slope from regions of negative slope. Indicate where the slope magnitude is large versus small.
(c) A solution curve passes through the point (0,1). Using only your slope field and qualitative reasoning (no algebraic solution required), predict the concavity of this solution for small positive values of x and justify your answer.