Use the Euler method with $h=0,1$ and $h=0,05$ to estimate the solution at $t =1$ for the IVP

$$\begin{array}{l}{x^{\prime}=x+y} \\ {y^{\prime}=y+z} \\ {z^{\prime}=-y+2 z}\end{array}$$

, where $x(0)=1, y(0)=1, z(0)=1$. Comment on the differences in results.

A 28-year-old is in the urgent care center after stepping on a rusty tent nail. The nurse evaluates the patient’s immunity status and notes that the patient thinks she had her last tetanus booster about 10 years ago, just before starting college. Which immunization would be most appropriate at this time?
a. Immunoglobulin intravenous (Gammar-P IV)
b. DTaP (Daptacel) (diphtheria, tetanus, and acellular pertussis)
c. Tdap (Adacel) (diphtheria, tetanus, and acellular pertussis)
d. No immunizations are necessary at this time.

A patient with a small bowel obstruction needs a central intravenous catheter for parenteral nutrition. Outline the steps you will take to assist the primary provider during insertion of the central line, including evidence-based infection prevention measures.

If air resistance is proportional to the square of the instantaneous velocity, then the velocity v of a mass m dropped from a given height h is determined from $m \frac{d v}{d t}=m g-k v^{2}, k>0$. Let $v(0)=0, k=0.125, m=5$ slugs, and $g=32 \mathrm{ft} / \mathrm{s}^{2}$. (a) Use the RK4 method with h=1 to approximate the velocity v(5). (b) Use a numerical solver to graph the solution of the IVP on the interval [0, 6]. (c) Use separation of variables to solve the IVP and then find the actual value v(5).

$(a)$ Does Picard's Theorem apply to the given IVP ? Explain. $(b)$ If your answer to part $(a)$ is yes, is there a largest rectangle for which Picard's condition s hold ? $(c)$If your answer to part $(a)$ is no, are there other initial conditions $y( t_{0}) = y_{0}$ for which the answer would be yes? Try describe the set of all such points. $y^{\prime}=\frac{t-y}{t+y}, \quad y(0)=-1$

$y=1 /\left(x^{2}+c\right)$ is a one-parameter family of solutions of the first-order $\text { DE } y^{\prime}+2 x y^{2}=0$. Find a solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined. $y\left(\frac{1}{2}\right)=-4$

Obtain solution for the IVP

$$\overrightarrow{\textbf{x}}'=\begin{bmatrix}1&1&0\\1&1&0\\0&0&-1\end{bmatrix}\overrightarrow{\textbf{x}}, \overrightarrow{\textbf{x}}(0)=\begin{bmatrix}2\\4\\2\end{bmatrix}$$

.

Approximate the solution to the IVP at $t = 0.2$ by Richardson 's extrapolation, using Euler's method with $h = 0.1$ and $h / 2 = 0 .05$. Compare with exact solutions when possible. $y^{\prime}=t y, \quad y(0)=1$

Calculator Again Consider the IVP $y^{'} = ty$, $y(0) = 1$. $(a)$ Use Euler's method to approximate the solution at $r = 1$ with step sizes $l , 1/2, l/4$, and $1/8$. $(b)$ Solve the problem exactly , and compare the result at $t = 1$ with the approximations calculated in part $(a)$ ,

Discuss why it is technically incorrect to say that the function

$$y = \left\{ \begin{array} { l l } { 1 - e ^ { - x } , } & { 0 \leq x \leq 1 } \\ { ( e - 1 ) e ^ { - x } , } & { x > 1 } \end{array} \right.$$

is a solution of the IVP

$$\frac { d y } { d x } + y = f ( x ) , y ( 0 ) = 0 \quad \text { where } \quad f ( x ) = \left\{ \begin{array} { l l } { 1 , } & { 0 \leq x \leq 1 } \\ { 0 , } & { x > 1 } \end{array} \right.$$

on the interval $[0, \infty )$.

Solve the IVP

$$\overrightarrow{\textbf{x}}'=\begin{bmatrix}2&0\\0&3\end{bmatrix}\overrightarrow{\textbf{x}},\overrightarrow{\textbf{x}}(0)=\begin{bmatrix}5\\4\end{bmatrix}$$

. Sketch the trajectory.

Solve the IVP

$$\overrightarrow{\textbf{x}}'=\begin{bmatrix}-2&4\\1&1\end{bmatrix}\overrightarrow{\textbf{x}},\overrightarrow{\textbf{x}}(0)=\begin{bmatrix}-1\\1\end{bmatrix}$$

. Sketch the trajectory.

Using the Laplace transform and showing the details of your work, solve the IVP: y''1=-2y1+2y2, y''2=2y1-5y2, y1(0)=1, y'1(0)=0, y2(0)=3, y'2(0)=0

When chemotherapy with alkylating drugs is planned, the nurse expects to implement which intervention to prevent nephrotoxicity?
a. Hydrating the patient with intravenous fluids before chemotherapy
b. Limiting fluids before chemotherapy
c. Monitoring drug levels during chemotherapy
d. Assessing creatinine clearance during chemotherapy