Find the general solution for each differential equation. $x \ln x \frac{d y}{d x}+y=2 x^{2}$

Find the value of each integral that converges. $\int_{4}^{\infty} \ln (5 x) d x$

How can you classify the results of operations on real numbers?

Use the integration feature on a graphing calculator and successively larger values of b to estimate $\int_{0}^{\infty} f(x) d x$. $\int_{0}^{b} e^{-x} \cos x d x$

Determine whether the matrix is orthogonally diagonalizable.

$$\left[\begin{array}{rrr} {5} & {-3} & {8} \\ {-3} & {-3} & {-3} \\ {8} & {-3} & {8} \end{array}\right]$$

From the following data, find the energy required to dissociate a KCl molecule into a K atom and a Cl atom. The first ionization potential of K is 4.34 eV; the electron affinity of Cl is 3.82 eV; the equilibrium separation of KCl is 2.79 A. (Hint: Show that the mutual potential energy of $\mathrm{K}^{+} \text {and } \mathrm{Cl}^{-}$ is $-(14.40 / R) \mathrm{eV}$ if R is given in Angstroms).

Use the rules for derivatives to find the derivative of each function defined as follows. $g(t)=t^{3}\left(t^{4}+5\right)^{7 / 2}$

Why is it unnecessary to find the value of $\lambda$ when using the method explained in

Kate has five sandwiches to share with three of her friends. If each person gets the same amount of sandwich, how much will each person get?

Write the recursive formula for the geometric sequence. 2, 5, 12.5, 31.25, 78.125, $\ldots$

Find the derivative of the function. $f(z)=\frac{1}{z^{2}+1}$

Solve for X in the equation, given the matrices

$$A=\begin{bmatrix} -2 & -1 \\ 1 & 0 \\ 3 & -4 \end{bmatrix}$$

and

$$B=\begin{bmatrix} 0 & 3 \\ 2 & 0 \\ -4& -1 \end{bmatrix}$$

,

Evaluate $X=3A-2B$.

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The matrix for a linear transformation A relative to the basis B is equal to the product $P A^{\prime} P^{-1},$ where P is the transition matrix from B' to B, A' is the matrix for the linear transformation relative to basis B' , and $P ^{- 1}$ is the transition matrix from B to B'. (b) The standard basis for $R^n$ will always make the coordinate matrix for the linear transformation T the simple st matrix possible.

Solve the absolute value equation. |2x - 8| = 16