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Magazine Problem 67
One of the key ideas for sketching solutions from vector field plots is that a solution curve must be monotonic, that is, \(x(t)\) is either increasing or decreasing or constant but cannot switch from one behavior to another. We showed that a solution \(x(t)\) could not start by increasing and then switch to decreasing. Suppose that \(x(t)\) is a solution of the differential equation \(\frac{d x}{d t}=g(x)\) and that \(x(t)\) starts off decreasing with time. Show that \(x(t)\) cannot switch to increasing.
Problem 77
For Problems \(77-88\) find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d y}{d t}=2-3 y $$
Problem 78
Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d y}{d t}=y-2 $$
Problem 79
Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d y}{d t}=y(2-y)(y-3) $$
Problem 80
Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d y}{d t}=y(y-1)(y-2) $$
Problem 81
Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d N}{d t}=N \ln \left(\frac{2}{N}\right) \quad N>0 $$
Problem 82
Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d N}{d t}=\frac{N-1}{N+1} \quad N \geq 0 $$
Problem 83
Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable.$$ \frac{d y}{d x}=\frac{y^{2}-y}{y^{2}+1} $$
Problem 84
Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d y}{d x}=\frac{1}{y^{3}}-\frac{1}{y} \quad, \quad y>0 $$
Problem 85
Find all equilibria, and, by calculating the eigenvalue of the differential equation, determine which equilibria are stable and which are unstable. $$ \frac{d x}{d t}=x e^{-x} $$
