Chapter 5

In Problems 59-72, solve the initial-value problem. $$ \frac{d W}{d t}=e^{-3 t}, \text { for } t \geq 0 \text { with } W(0)=2 / 3 $$

In Problems 59-72, solve the initial-value problem. $$ \frac{d W}{d t}=e^{-5 t}, \text { for } t \geq 0 \text { with } W(0)=1 $$

In Problems 59-72, solve the initial-value problem. $$ \frac{d T}{d t}=\sin (\pi t), \text { for } t \geq 0 \text { with } T(0)=3 $$

In Problems 59-72, solve the initial-value problem. $$ \frac{d T}{d t}=\cos (\pi t), \text { for } t \geq 0 \text { with } T(0)=3 $$

In Problems 59-72, solve the initial-value problem. $$ \frac{d y}{d x}=\frac{e^{-x}+e^{x}}{2}, \text { for } x \geq 0 \text { with } y=0 \text { when } x=0 $$

Suppose that the length of a certain organism at age \(x\) is given by \(L(x)\), which satisfies the differential equation $$ \frac{d L}{d x}=e^{-0.1 x}, \quad x \geq 0 $$ Find \(L(x)\) if the limiting length \(L_{\infty}\) is given by $$ L_{\infty}=\lim _{x \rightarrow \infty} L(x)=25 $$ How big is the organism at age \(x=0\) ?

Fish are indeterminate growers; that is, their length \(L(x)\) increases with age \(x\) throughout their lifetime. If we plot the growth rate \(d L / d x\) versus age \(x\) on semilog paper, a straight line with negative slope results. Set up a differential equation that relates growth rate and age. Solve this equation under the assumption that \(L(0)=5, L(1)=10\), and $$ \lim _{x \rightarrow \infty} L(x)=20 $$ Graph the solution \(L(x)\) as a function of \(x\).

An object is dropped from a height of \(100 \mathrm{ft}\). Its acceleration is \(32 \mathrm{ft} / \mathrm{s}^{2}\). When will the object hit the ground, and what will its speed be at impact?

Suppose that the growth rate of a population at time \(t\) undergoes seasonal fluctuations according to $$ \frac{d N}{d t}=3 \sin (2 \pi t) $$ where \(t\) is measured in years and \(N(t)\) denotes the size of the population at time \(t .\) If \(N(0)=10\) (measured in thousands), find an expression for \(N(t) .\) How are the seasonal fluctuations in the growth rate reflected in the population size?

Suppose that the amount of water contained in a plant at time \(t\) is denoted by \(V(t) .\) Due to evaporation, \(V(t)\) changes over time. Suppose that the change in volume at time \(t\), measured over a 24-hour period, is proportional to \(t(24-t)\), measured in grams per hour. To offset the water loss, you water the plant at a constant rate of 4 grams of water per hour. (a) Explain why $$ \frac{d V}{d t}=-a t(24-t)+4 $$ \(0 \leq t \leq 24\), for some positive constant \(a\), describes this situation. (b) Determine the constant \(a\) for which the net water loss over a 24 -hour period is equal to 0 .