Find G(s) such that the Laplace transform of the charge on the capacitor, Q(s) = L{q(t)}, can be expressed as Q(s) = G(s)E(s), where E(s) = L{ e(t)} is the Laplace transform of the impressed voltage. Assume that at time t = 0 the charge on the capacitor is zero and the currents $i_{1} \text { and } i_{2}$ are zero.

Use an inverse matrix to solve each system of linear equations or matrix equation.

$$\begin{array}{l} {-x_{1}+x_{2}+2 x_{3}=1} \\ {2 x_{1}+3 x_{2}+x_{3}=-2} \\ {5 x_{1}+4 x_{2}+2 x_{3}=4} \end{array}$$

Prove that a rotation of $\theta$, where $\cot 2 \theta=(a-c) / b$, will eliminate the xy-term from the equation $a x^{2}+b x y+c y^{2}+d x+e y+f=0$.

Perform the operations when

$A=$

$$\begin{bmatrix} 1 & 2 \\ 0 & -1 \\ \end{bmatrix}$$

.

Evaluate

$$AI$$

Find the products $AB$ and $BA$ for the diagonal matrices.

$A=$

$$\begin{bmatrix} 3&0&0\\ 0&-5&0\\ 0&0&0 \end{bmatrix}$$

$B=$

$$\begin{bmatrix} -7&0 &0\\ 0&4 &0 \\ 0&0 &12 \end{bmatrix}$$

Solve the system of first-order linear differential equations.

$$\begin{aligned} &y_{1}^{\prime}=y_{1}-y_{2}\\ &y_{2}^{\prime}=2 y_{1}+4 y_{2} \end{aligned}$$

Differentiate the function. $f(t)=\frac{1}{2} t^{6}-3 t^{4}+t$

What geometrical shape are the equipotential surfaces between two charged parallel plates?

Is $-\frac{1}{2}n+\frac{1}{2}n+3-n$ equivalent to 3? Explain.

Find the domain and range of the function represented by the graph.

A 0.20-kg glass cup at $20^{\circ} \mathrm{C}$ is filled with 0.40 kg of hot water at $90^{\circ} \mathrm{C}.$ Neglecting any heat losses to the environment, what is the equilibrium temperature of the water?

Use formal substitution to find the indefinite integral. Check your result by differentiating. $\int t \sqrt{t^{2}+1} d t$

Two coils are mutually coupled, with $L_{1}=50 \mathrm{mH}$, $L_{2}=120 \mathrm{mH},$ and $k=0.5$. Calculate the maximum possible equivalent inductance if: (a) the two coils are connected in series (b) the coils are connected in parallel

Sketch the graph of the amount of a particular brand of coffee sold by a store as a function of the price of the coffee.