For each of the following differential equations in Problem, apply the method of Frobenius to obtain the solution. $x^{2} y^{\prime \prime}+\left(x^{2}-x\right) y^{\prime}+2 y=0$

Create a table of values for the function.

To get more than an infinitesimal amount of work out of a Carnot engine, we would have to keep the temperature of its working substance below that of the hot reservoir and above that of the cold reservoir by non-infinitesimal amounts. Consider, then, a Carnot cycle in which the working substance is at temperature $T_{h w}$ as it absorbs heat from the hot reservoir, and at temperature $T_{c w}$ as it expels heat to the cold reservoir. Under most circumstances the rates of heat transfer will be directly proportional to the temperature differences: $\frac{Q_{h}}{\Delta t}=K\left(T_{h}-T_{h w}\right)$ and $\frac{Q_{c}}{\Delta t}=K\left(T_{c w}-T_{c}\right).$ I've assumed here for simplicity that the constants of proportionality (K) are the same for both of these processes. Let us also assume that both processes take the same amount of time, so the $\Delta t's$ are the same in both of these equations. (a) Assuming that no new entropy is created during the cycle except during the two heat transfer processes, derive an equation that relates the four temperatures $T_{h}, T_{c}, T_{h w},$ and $T_{c w}.$ (b) Assuming that the time required for the two adiabatic steps is negligible, write down an expression for the power (work per unit time) output of this engine. Use the first and second laws to write the power entirely in terms of the four temperatures (and the constant K), then eliminate $T_{c w}$ using the result of part (a). (c) When the cost of building an engine is much greater than the cost of fuel (as is often the case), it is desirable to optimize the engine for maximum power output, not maximum efficiency. Show that, for fixed $T_{h}$ and $T_{c}$ the expression you found in part (b) has a maximum value at $T_{h w}=\frac{1}{2}\left(T_{h}+\sqrt{T_{h} T_{c}}\right).$ Find the corresponding expression for $T_{c w}.$ (d) Show that the efficiency of this engine is $1-\sqrt{T_{c} / T_{h}}.$ Evaluate this efficiency numerically for a typical coal-fired steam turbine with $T_{h}=600^{\circ} \mathrm{C}$ and $T_{c}=25^{\circ} \mathrm{C},$ and compare to the ideal Carnot efficiency for this temperature range. Which value is closer to the actual efficiency, about 40% of a real coal-burning power plant?

Apply least squares regression to the data for the planets Mercury, Earth, Jupiter, and Uranus to develop a model to predict orbital period from distance.

Use Laplace transforms to solve the initial-value problem. $x^{\prime \prime}-8 x^{\prime}+15 x=5 t e^{2 t}, \ x(0)=5, \ x^{\prime}(0)=4$

Sperm are produced in the ________. a. scrotum, b. seminal vesicles, c. seminiferous tubules, d. prostate gland.

A normal distribution has a standard deviation of 1. We want to verify a claim that the mean is greater than 12. A sample of 36 is taken with a sample mean of 12.5.

$$\begin{aligned} &H 0: \mu \leq 12\\ &H_{a}: \mu>12 \end{aligned}$$

The p-value is 0.0013 Draw a graph that shows the p-value.

Registered nurses earned an average annual salary of $69,110. For that same year, a survey was conducted of 41 California registered nurses to determine if the annual salary is higher than$69,110 for California nurses. The sample average was $71,121 with a sample standard deviation of$7,489. Conduct a hypothesis test.

Many endemic species are found in areas that are geographically isolated. Suggest a plausible scientific explanation for why this is so.

Explain two different excretory systems other than the kidneys.

Which of the following statements about biomes is false? a. Chaparral is dominated by shrubs. b. Savannas and temperate grasslands are dominated by grasses. c. Boreal forests are dominated by deciduous trees. d. Lichens are common in the arctic tundra.

How does the polygastric digestive system aid in digesting roughage?

Let A and B be sets. Prove $A \subseteq B$ iff $\mathscr{P}(A) \subseteq \mathscr{P}(B)$.

Define $T \in \mathcal{L}\left(\mathbf{C}^{2}\right)$ by $T(w, z)=(-z, w).$ Find the generalized eigenspaces corresponding to the distinct eigenvalues of T.