91Ó°ÊÓ

Decide whether the statement is true or false. Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=2 x-y .\) Justify your answer. $$\text { If } y=f(x), \text { then } d^{2} y / d x^{2}=2-(2 x-y)$$

Solve the differential equations in Problems \(34-43 .\) Assume \(a, b,\) and \(k\) are nonzero constants. $$\frac{d R}{d t}=a R+b$$

Are the statements true or false? Give an explanation for your answer. The system of differential equations \(d x / d t=-x+x y^{2}\) and \(d y / d t=y-x^{2} y\) requires initial conditions for both \(x(0)\) and \(y(0)\) to determine a unique solution.

Decide whether the statement is true or false. Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=2 x-y .\) Justify your answer. If \(f(1)=5,\) then (1,5) could be a critical point of \(f\)

Is the statement true or false? Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=g(x) .\) If the statement is true, explain how you know. If the statement is false, give a counterexample. If \(g(x)\) is increasing for all \(x,\) then the graph of \(f\) is concave up for all \(x.\)

Solve the differential equations in Problems \(34-43 .\) Assume \(a, b,\) and \(k\) are nonzero constants. $$\frac{d y}{d t}=k y^{2}\left(1+t^{2}\right)$$

Federal or state agencies control hunting and fishing by setting a quota on how many animals can be harvested each season. Determining the appropriate quota means achieving a balance between environmental concerns and the interests of hunters and fishers. For example, when a June 8,2007 decision by the Delaware Superior Court invalidated a two-year moratorium on catching horseshoe crabs, the Delaware Department of Natural 91Ó°ÊÓ and Environmental Control imposed instead an annual quota of 100,000 on male horseshoe crabs. Environmentalists argued this would exacerbate a decrease in the protected Red Knot bird population that depends on the crab for food. For a population \(P\) that satisfies the logistic model with harvesting, $$\frac{d P}{d t}=k P\left(1-\frac{P}{L}\right)-H$$ show that the quota, \(H,\) must satisfy \(H \leq k L / 4,\) or else the population \(P\) may die out. (In fact, \(H\) should be kept much less than \(k L / 4 \text { to be safe. })\)

Is the statement true or false? Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=g(x) .\) If the statement is true, explain how you know. If the statement is false, give a counterexample. If \(g(x)\) is increasing for \(x>0,\) then so is \(f(x).\)

Decide whether the statement is true or false. Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=2 x-y .\) Justify your answer. The graph of \(f\) is decreasing whenever it lies above the line \(y=2 x\) and is increasing whenever it lies below the line \(y=2 x\)

Explain what is wrong with the statement. The differential equation \(d P / d t=0.08 P-0.0032 P^{2}\) has one equilibrium solution, at \(P=25\)