Solve the given system of differential equations.

$$x _ { 1 } ^ { \prime } = 2 x _ { 1 } , \quad x _ { 2 } ^ { \prime } = x _ { 2 } - x _ { 3 } , \quad x _ { 3 } ^ { \prime } = x _ { 2 } + x _ { 3 }.$$

A parallel-plate capacitor is fully charged and connected to an inductor. The capacitor initially stores charge q and electric potential energy $U ^ { E }$. Once the circuit is closed, charge carriers flow from the capacitor, producing a current in the circuit. When the charge on the capacitor has been reduced to q/2, what is the magnetic potential energy $U ^ { B }$ stored in the inductor? Report your answer in terms of $U ^ { E }$.

Find the points at which the following planes intersect the coordinate axes and find equations of the lines where the planes intersect the coordinate planes. Sketch a graph of the plane. x+3y-5z-30=0

A model for a company's revenue is $R=-15 p^{2}+300 p+12,000,$ where p is the price in dollars of the company's product. What price will maximize revenue? Find the maximum revenue.

Fill the correct vocabulary term for the given sentence: The triangular faces of a pyramid that are not bases are ____________________ faces.

Write the fraction as a percent.

$$\frac{3}{5}$$

Evaluate the given expression. Take

$$A=\left[\begin{array}{rr}{0} & {-1} \\ {1} & {0} \\ {-1} & {2}\end{array}\right]$$

,

$$B=\left[\begin{array}{rr}{0.25} & {-1} \\ {0} & {0.5} \\ {-1} & {3}\end{array}\right]$$

, and

$$C=\left[\begin{array}{rr}{1} & {-1} \\ {1} & {1} \\ {-1} & {-1}\end{array}\right]$$

. 2A-C

Briefly explain why glass–ceramics may not be transparent.

Completely factor the expression. $5-x+5 x^{2}-x^{3}$

Determine whether each equation defines y as a function of x.

$$x + y ^ { 3 } = 27$$

For the following differential equation

$$\begin{gather*} x \ \dfrac{dy}{dx} = -\dfrac{1}{\ln x}, \end{gather*}$$

(a) List into which the equation falls: autonomous, separable, linear, exact, Bernoulli, homogeneous. (b) Solve the equation. If it has more than one classification, solve it two different ways.

Using the Clapeyron equation, estimate the enthalpy of vaporization of water at 300 kPa, and compare it to the tabulated value.

Use variation of parameters to solve the given system.

$$\mathbf{X}^{\prime}=\left(\begin{array}{rr}{1} & {-1} \\ {1} & {1}\end{array}\right) \mathbf{X}+\left(\begin{array}{l}{3} \\ {3}\end{array}\right) e^{t}$$

Use a graphing utility to graph the equation. Use a standard viewing window. Approximate any x- or y-intercepts of the graph. $2 y-x^{2}+8=2 x$