Third-Order Examples: For each of the nonhomogeneous linear DEs in $(a)$ Ver(fy that the given $y_{1}, y_{2}, y_{3}$ satisfy the corresponding homogeneous equation. $(b)$Use the Superposition Principle, with appropriate coefficients, to state the general solution $y_{1}(t)$ to the corresponding homogeneous equation. $(c)$ Verify that the given $y_{p}(t)$ is a particular solution to the given nonhomogeneous s DE. $(d)$ Use the Nonhomogeneous Principle to write the general solution $y(t)$ to the nonhomogeneous DE. $(e)$ Solve the NP consisting of the nonhomogeneous DE and the given initial conditions

$$\begin{array}{l}{y^{\prime \prime \prime}+y^{\prime \prime}-y^{\prime}-y=4 \sin t+3} \\ {y_{1}=e^{t}, y_{2}=e^{-t} \cdot y_{3}=t e^{-t}} \\ {y_{p}=\cos t-\sin t-3} \\ {y(0)=1 . y^{\prime}(0)=2, \quad y^{\prime \prime}(0)=3}\end{array}$$

Use double integration to find the center of mass of a lamina covering the given region in the plane and having the specified density $\rho$.

$$\begin{array}{l}{\rho(x, y)=3 x \text { over the region bounded by } y=0,} \\ {y=x^{2}, \text { and } x=6}\end{array}$$

A long, closed cylindrical tank contains a diatomic gas that is maintained at a uniform temperature that can be varied. When you measure the speed of sound in the gas as a function of the temperature of the gas, you obtain these results:

$$\begin{matrix}\mathrm{T\left(^{\circ} \mathbf{C}\right)} & \text{-20.0} & \text{0.0} & \text{20.0} & \text{40.0} & \text{60.0} & \text{80.0}\\ \text{v m/s} & \text{324} & \text{337} & \text{349} & \text{361} & \text{372} & \text{383}\\ \end{matrix}$$

(a) Explain how you can plot these results so that the graph will be well fit by a straight line. Construct this graph and verify that the plotted points do lie close to a straight line.

(b) Because the gas is diatomic, $\gamma =1.40$. Use the slope of the line in part (a) to calculate M, the molar mass of the gas. Express M in grams/mole. What type of gas is in the tank?

Decide whether Rolle's theorem can be applied to f on the interval indicated.

$$f(x)=3 x+\sec x \text{on}[-\pi, \pi]$$

Evaluate the expression. $5^{\log _{5} 12}$

Identify which of these types of sampling is used: random, systematic, convenience, stratified, or cluster. The sexuality of women was discussed in Shere Rite's book Women and Love: A Cultural Revolution. Her conclusions were based on sample data that consisted of 4500 mailed responses from 100,000 questionnaires that were sent to women.

Solve the equation by graphing.

$$x ^ { 2 } - 4 x = 5$$

Another series of nuclear reactions that can produce energy in the interior of stars is the cycle described below. This cycle is most efficient when the central temperature in a star is above $1.6 \times 10 ^ { 7 }$ K. Because the temperature at the center of the Sun is only $1.5 \times 10 ^ { 7 }$ K, the following cycle produces less than 10% of the Sun’s energy. Nucleus D decays through positron emission to produce nucleus E. Identify nucleus E.

How can you classify triangles using angles? using sides?

By substituting $e^{\ln 2}$ for 2 in $y=2^{x}$, find $\frac{d y}{d x}$.

Why was corn an important crop to early peoples?

Estimate the given percent. $35 \%$ of $147 \approx$

Use substitution to determine whether the given linear expression is a factor of the given polynomial. $2 x^{3}+10 x^{2}-28 x ; x+7$

A stereo speaker is placed between two observers who are 36 m apart, along the line connecting them. If one observer records an intensity level of 60 dB, and the other records an intensity level of 80 dB, how far is the speaker from each observer?