MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations
Problem
Two autonomous differential equations are given.
(I) y′=0.05y
(II) y′=0.05y(100−y)
For each differential equation, imagine its slope field and the family of solution curves. No explicit solutions are required.
(1) Describe two qualitative differences you would expect to see between the slope fields of (I) and (II).
(2) Compare the long-term behavior of solutions to (I) and (II) that start with the same initial value y(0)=10. Explain your reasoning using features visible in their respective slope fields.
(3) For differential equation (II), indicate the interval(s) of y-values on which solution curves are concave up and the interval(s) on which they are concave down, and justify your answer without computing second derivatives.