Show that the given signal is a solution of the difference equation. Then find the general solution of the difference equation. $y_{k}=k-2 ; y_{k+2}-4 y_{k}=8-3 k$

A drug is carried into an organ of volume V $cm^3$ by a liquid that enters the organ at the rate of a $\mathrm{cm}^{3} / \mathrm{sec}$ and leaves it at the rate of b $\mathrm{cm}^{3} / \mathrm{sec}$. The concentration of the drug in the liquid entering the organ is c $\mathrm{g} / \mathrm{cm}^{3}$. If the concentration of the drug in the organ at time t is increasing at the rate of $x^{\prime}(t)=\frac{1}{V}\left(a c-b x_{0}\right) e^{-b t v}$ $\mathrm{g} / \mathrm{cm}^{3} / \mathrm{sec}$ and the concentration of the drug in the organ initially is $x_{0} \mathrm{g} / \mathrm{cm}^{3}$, show that the concentration of the drug in the organ at time t is given by $x(t)=\frac{a c}{b}+\left(x_{0}-\frac{a c}{b}\right) e^{-b c v}$

Show that the given signal is a solution of the difference equation. Then find the general solution of that difference equation. $y_{k}=1+k ; y_{k+2}-8 y_{k+1}+15 y_{k}=2+8 k$.

Show that the given signal is a solution of the difference equation. Then find the general solution of the difference equation. $y_{k}=1+2 k ; y_{k+2}-25 y_{k}=-48 k-20$

Show that the given signal is a solution of the difference equation. Then find the general solution of that difference equation.

$$y_k=2-2 k ; y_{k+2}-\frac{9}{2} y_{k+1}+2 y_k=3 k+2$$

Show that the given signal is a solution of the difference equation. Then find the general solution of that difference equation.

$$y_k=k^2 ; y_{k+2}+3 y_{k+1}-4 y_k=10 k+7$$

Find a basis for the solution space of the difference equation. Prove that the solutions you find span the solution set.

$$16 y_{k+2}+8 y_{k+1}-3 y_k=0$$

Find a basis for the solution space of the difference equation. Prove that the solutions you find span the solution set. $6 y_{k+2}+y_{k+1}-2 y_{k}=0$

Find a basis for the solution space of the difference equation. Prove that the solutions you find span the solution set.

$$y_{k+2}-7 y_{k+1}+12 y_k=0$$

Let V and W be vector spaces, let $T : V \rightarrow W$ be a linear transformation, and let $\left\{\mathbf{v}_{1}, \dots, \mathbf{v}_{p}\right\}$ be a subset of V. Show that if $\left\{\mathbf{v}_{1}, \dots, \mathbf{v}_{p}\right\}$, is linearly dependent in W. This fact shows that if a linear transformation maps a set $\left\{\mathbf{v}_{1}, \dots, \mathbf{v}_{p}\right\}$ onto a linearly independent set $\left\{T\left(\mathbf{v}_{1}\right), \ldots, T\left(\mathbf{v}_{p}\right)\right\}$, then the original set is linearly independent, too (because it cannot be linearly dependent).

Using z-transform methods, solve the following difference equations: (a) $6 y_{k+2}+y_{k+1}-y_{k}=3$ subject to $y_{0}=y_{1}=0$ (b) $y_{k+2}-5 y_{k+1}+6 y_{k}=5$ subject to $y_{0}=0, y_{1}=1$ (c) $y_{n+2}-5 y_{n+1}+6 y_{n}=\left(\frac{1}{2}\right)^{n}$ subject to $y_{0}=y_{1}=0$ (d) $y_{n+2}-3 y_{n+1}+3 y_{n}=1$ subject to $y_{0}= 1, y_{1}=0$ (e) $2 y_{n+2}-3 y_{n+1}-2 y_{n}=6 n+1$ subject to $y_{0}= 1, y_{1}=2$ (f) $y_{n+2}-4 y_{n}=3 n-5$ subject to $y_{0}=y_{1}=0$

Let V and W be vector spaces, and let $T : V \rightarrow W$ be a linear transformation. Given a subspace U of V, let T(U) denote the set of all images of the form T(x), where x is in U. Show that T(U) is a subspace of W.

Assume the signals listed are solutions of the given difference equation. Determine if the signals form a basis for the solution space of the equation. Justify your answers using appropriate theorems. $1^{k}, 3^{k} \cos \frac{k \pi}{2}, 3^{k} \sin \frac{k \pi}{2} ; y_{k+3}-y_{k+2}+9 y_{k+1}-9 y_{k}=0$.

Find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability. $P(z < -1.22$ or $z > 1.32)$