For a matrix $m=\begin{pmatrix}a & b \\ c & d\end{pmatrix}$ let $\delta_m=ad-bc$.

1. Show that for $m, m'\in \mathbb{R}^{2\times 2}$ and $\lambda \in \mathbb{R}$ it holds that $\delta_{mm'}=\delta_m\delta_{m'}$ and $\delta_{\lambda m}=\lambda^2\delta_m$.

2. Show that the following statements for a matrix $m\in \mathbb{R}^{2\times 2}$ are equivalent:

(i) $\delta_m\neq 0$

(ii) the matrix $m$ is invertible.

3. Let $m, m'\in \mathbb{R}^{2\times 2}$ be invertible...

Show equivalence: Invertible matrix]]>

We have the matrices \begin{equation*}s:=\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}, \ d:=\frac{1}{2}\begin{pmatrix}-1 & -\sqrt{3} \\ \sqrt{3} & -1\end{pmatrix}\end{equation*} and the points \begin{equation*}p:=\begin{pmatrix}2 \\ 0 \end{pmatrix}, \ q:=\begin{pmatrix}-1 \\ \sqrt{3} \end{pmatrix}, \ r:=\begin{pmatrix}-1 \\ -\sqrt{3} \end{pmatrix}\end{equation*}

I draw the points $p, q, r$ and calculate also the points $sp, sq, sr, dp, dq, dr$ and I noticed that $s$ is a...

Set is closed as for multiplication of matrices]]>

We have the following iteration from Newton's method \begin{align*}x_{k+1}&=x_k-\frac{f(x_k)}{f'(x_k)}=x_k-\frac{x_k^n-a}{nx_k^{n-1}}=\frac{x_k\cdot nx_k^{n-1}-\left (x_k^n-a\right )}{nx_k^{n-1}}=\frac{ nx_k^{n}-x_k^n+a}{nx_k^{n-1}}\\ & =\frac{ (n-1)x_k^{n}+a}{nx_k^{n-1}}\end{align*}

I want to show that for $x_0\geq a^{1/n}$ the method converges monotonically to the root.

So first we have to show that the sequence $(x_k)$ is monotone decreasing, right?

I have done the...

Monotonically convergence to the root]]>

I saw the below sentence in some notes:

Let $A\in \mathbb{R}^{n\times n}$ be a not necessarily symmetric, strictly positive definite matrix, $x^TAx>0$, $x\neq 0$ und $Q\in \mathbb{R}^{n\times n}$ an orthogonal matrix, then $B=Q^TAQ$ has a LU decomposition.

I want to understand this implication, so I thought that this could hold for the following reason:

Since the matrix $A$ is strictly positive definite matrix, we know that the determinant is always positiv, i.e. not zero...

B=Q^TAQ has a LU decomposition]]>

$\dfrac{|a+b|}{1+|a+b|}$ $\leq \dfrac{|a|}{1+|a|}$ +$\dfrac{|b|}{1+|b|}$]]>

So here is what I have so far:

The derivative is 1-ln(x)/x^2

Critical points are (e,1/e)

No concavity

Local max is also (e,1/e) (no local min)

no inflection points

Increase on (0, e) and decrease on (e, positive infinity)

Is this correct? I tried to do a graph to justify my work]]>

I have to find:

g(1)=

and

g(5)=

I have drawn the graph and I am a little unsure where to go from there. I know area is involved somehow but not entirely sure what to do. Any help is appreciated]]>

A. (3, 0)

B. (3 1/4, 0)

C. (3 3/4, 0)

D. (4 1/2, 0)

E. (5, 0)

What I did:

f(x) = AP + PB =\(\displaystyle \sqrt{5^2+x^2}+(10-x)=\sqrt{25+x^2}+10-x\)

In order to make AP + PB minimum, so:

f'(x) = 0

\(\displaystyle \frac12(25+x^2)^{-\frac12}(2x)+(-1)=0\)

\(\displaystyle \frac{x}{\sqrt{25+x^2}}=1\)...

[ASK] Minimum Length of AP + PB]]>

The cost of an individual ticket is \$14, and the cost of a season pass is \$175.

The season pass will admit Armin to any home basketball game at no additional cost.

What is the minimum number of home basketball games Armin must attend this season

in order for the cost of a season pass to be less than the total cost of buying an individual ticket for each game he attends...

-act.63 cost per game with season pass]]>

We have the iteration formula $$x_{j+1}=2x_j-3x_j^2$$ and using the starting point $x_0=0.3$ we get the following approximations for $\frac{1}{3}$ :

\begin{align*}&x_1=2x_0-3x_0^2=2\cdot 0.3-3\cdot 0.3^2=0.33 \\ &x_2=2x_1-3x_1^2=2\cdot 0.33-3\cdot 0.33^2=0.3333 \\ &x_3=2x_2-3x_2^2=2\cdot 0.3333-3\cdot 0.3333^2=0.33333333\end{align*}

Now I want to show that for $j\geq 1$ the first $2^j$ digits of $x_j$ are correct for $\frac{1}{3}$.

It is...

For j >1 the first 2^j digits are correct]]>

A. 4 square dm

B. 6 square dm

C. 8 square dm

D. 10 square dm

E. 12 square dm

What I've done:

\(\displaystyle x^2t=2βt=\frac{2}{x^2}\)

The production cost per unit of its top and its...

[ASK] Minimum Surface Area]]>

If we want to calculate the nodes $x_1, x_2$ and the weight functions $w_1, w_2$ for the Gaussian quadrature of the integral $$\int_{-1}^1f(x)\, dx\approx \sum_{j=1}^2w_jf(x_j)$$ is there a criteria that we have to consider at chosing the weight functions? I mean if we use e.g. Gauss-Legendre or Tschebyscheff-Jacobi? ]]>

For which values of $a\neq 0$ can we solve iteratively $ax^3-x-2=0$ by $x_{n+1}=ax_n^3-2, \ n=1,2, \ldots $ with appropriate $x_1$ ?

I have done the following:

$$ax^3-x-2=0 \Rightarrow x=ax^3-2$$ So we can consider a fix point problem with $\phi (x)=ax^3-2$.

So we have to check when the fix point iteration converges right?

For that $\phi$ has to be Lipshitz continuous, or not? ]]>

-----

Let $x,\,y,\,z\in [0,\,1]$. Find the maximum value of $\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

-----

For any natural number $n$, ($n\ge 3$), let $f(n)$ denote the number of non-congruent integer-sided triangles with perimeter $n$ (e.g., $f(3)=1,\,f(4)=0,\,f(7)=2$). Show that

a. $f(1999)>f(1996)$,

b. $f(2000)=f(1997)$.

-----

Remember to read the POTW submission guidelines to find out how to submit your...

Problem Of The Week #443 November 16th, 2020]]>

We consider the inner product $$\langle f,g\rangle:=\int_{-1}^1(1-x^2)f(x)g(x)\, dx$$ Calculate an orthonormal basis for the poynomials of degree maximum $2$.

I have applied the Gram-Schmidt algorithm as follows:

\begin{align*}\tilde{q}_1:=&1 \\ q_1:=&\frac{\tilde{q}_1}{\|\tilde{q}_1\|}=\frac{1}{\langle \tilde{q}_1, \tilde{q}_1\rangle}=\frac{1}{\int_{-1}^1(1-x^2)\cdot 1\cdot 1\, dx}=\frac{1}{\int_{-1}^1(1-x^2)\, dx}=\frac{1}{\left [x-\frac{x^3}{3}\right]_{-1}^1}=\frac{1}{\left...

Orthonormal basis for the poynomials of degree maximum 2]]>

Calculate using the Simpson's Rule the integral $\int_0^1\sqrt{1+x^4}\, dx$ approximately such that the error is less that $0,5\cdot 10^{-3}$. Which has to be $h$ ?

So we use here the composite Simpson's rule, right?

An upper bound of the error of that rule is defined as $$\frac{h^4}{180}(b-a)\max_{\xi \in [a,b]}|f^{(4)}(\xi)|$$ and we have to set this equal to $0,5\cdot 10^{-3}$, right?

So we get $$\frac{h^4}{180}\max_{\xi \in [a,b]}|f^{(4)}(\xi)|=0,5\cdot 10^{-3} $$

The...

Simpson's Rule]]>

Calculate the Cholesky decomposition of the matrix, the only non-vanishing elements are the diagonals $1,2,3, \lambda$ and all under and upper secondary diagonal elements are $1$.

For which $\lambda$ is the matrix singular?

Could you please explain the form of the Matrix?

Does the matrix on the diagonal have $1,2,3, \lambda$ ? So do we have a $4 \times 4$ matrix? Or are these the only non zero diagonal elements? ]]>

We consider the matrix $$A=\begin{pmatrix}1 & -2 & 5 & 0 & 5\\ 1 & 0 & 2 & 0 & 3\\ 1 & 2 & 5 & 4 & 6 \\ -2 & 2 & -4 & 1 & -6 \\ 3 & 4 & 9 & 5 & 11\end{pmatrix}$$ I want to find the LU decomposition.

How do we know if we have to do the decomposition with pivoting or without? ]]>

\(\displaystyle

\begin{cases}

y''(t)=-\frac{y(t)}{||y(t)||^3} \ , \forall t >0

\\

y(0)= \Big(\begin{matrix} 1\\0\end{matrix} \Big) \

\text{and}

\

y'(0)= \Big(\begin{matrix} 0\\1\end{matrix} \Big)

\end{cases}

\\

y(t) \in \mathbb{R}^2 \ \forall t

\)

Thanks in advance ^^]]>

At the cash register of a store we want to give change of worth $V$ cents of euro. Create and analyze a greedy algorithm that gives the change using the minimum number of coins.

Assume that the available coins are the euro subdivisions, i.e. $\{1, 2, 5, 10, 20, 50\}$ and that we have an unlimited number of coins for each subdivision.

For this question I have done the following pseudocode:

Code:

```
Algorithm min_number_coins (C, V)
Input: array C={1,2,5,10,20,50} in...
```

Data showed that 22% of people in a small town was infected with the COVID-19 virus. A random sample of six residents from this town was selected.

1) What is the probability that exactly two of these residents was infected?

2) What is the probability that at most 1 of these residents was infected?

Thank you]]>

If not, can someone walk me through the steps to get to the results that my professor got? Thank you.]]>

This seems to be rather common GRE Exam Question

Which appears to harder that what it is

It looks like a similar triagle solution

$\dfrac{800}{7000}=\dfrac{x}{1400}$]]>

I found the following algorithm for the

Why at the case else, we do not change the variable w to w+(W-w)/S

It is given that the negation of "$\forall a\in A: \alpha (a)$" is "$\exists a\in A: \neg \alpha (a)$" and the negation of "$\exists b\in B: \beta (b)$" is "$\forall b\in B: \neg \beta (b)$".

I want to show that the negation of "$\forall a \in A \ \exists b \in B \ : \ \alpha (a, b)$" is "$\exists a \in A \ \forall b \in B \ : \ \neg \alpha (a, b)$".

Do we show that as follows? \begin{align*}\neg \left (\forall a \in A \ \exists b \in B \ : \ \alpha (a, b)\right...

Logical negation of statements]]>

sqrt(x + 15) + sqrt(x) = 15

Any suggestions are appreciated how to approach the solution for this equation.]]>

$\sin \left(a-\dfrac{\pi}{6}\right)+\sin \left(b-\dfrac{\pi}{6}\right)+\sin \left(c-\dfrac{\pi}{6}\right)\ge 0$.]]>

\quad |\sin{\theta}\ge 1|$ is true for all and only the values of $\theta$ in which of the following sets

$a.\ \left\{-\dfrac{\pi}{2},\dfrac{\pi}{2}\right\}

\quad b.\ \left\{\dfrac{\pi}{2}\right\}

\quad c.\ \left\{\theta | -\dfrac{\pi}{2}< \theta < \dfrac{\pi}{2}\right\}

\quad d.\ \left\{\theta | -\dfrac{\pi}{2}\le \theta < \dfrac{\pi}{2} \le \right\}

\quad e.\ \textit{empty set}$

I chose b since $\sin\theta$ can only be 1 at...

act.59 sin theta >= 1]]>

ok kinda catchy but I chose B

$A =10\cdot 12 =120,\quad B=1\cdot2\cdot10\cdot12=240$

$\begin{array}{ll}

A &\textit{ Quantity A is greater}\\

B &\textit{ Quantity B is greater}\\

C &\textit{ The 2 Quantities are equal}\\

D &\textit{The relationship cannot be determined}

\end{array}$}

how would you rate this Easy,Mediam,Hard?

typos??]]>

can someone prove this and show the process in detail? Many thanks ]]>

Find dy/dx]]>

I want to solve the recurrence relation

$$T(n)=4T{\left( \frac{n}{3} \right)}+n \log{n}.$$

I thought to use the Master Theorem.

We have $a=4, b=3, f(n)=n \log{n}$.

$\log_b{a}=\log_3{4}$

$n^{\log_b{a}}=n^{\log_3{4}}$

How can we find a relation between $n^{\log_{3}{4}}$ and $n \log{n}$ ? ]]>

$t(t-4)y'+y=0 \quad y(2)=2\quad 0<t<4$

ok my first step was isolate y'

s

$y'=-\dfrac{y}{t(t-4)}$

not sure what direction to go since we are concerned about an interval]]>

I have calculated an approximation to $\frac{\pi}{2}$ using Newton's method on $f(x)=\cos (x)$ with starting value $1$. After 2 iterations we get $1,5707$.

Which conditions does the starting point has to satisfy so that the convergence of the sequence of the newton iterations to $\frac{\pi}{2}$ is guaranteed? ]]>

Show that the interpolation exercise for cubic splines with $s(x_0), s(x_1), , \ldots , s(x_m)$ at the points $x_0<x_1<\ldots <x_m$, together with one of $s'(x_0)$ or $s''(x_0)$ and $s'(x_m)$ or $s''(x_m)$ has exactly one solution.

Could you give me a hint how we could show that? Do wwe have to assume that we have two different solutions annd get a contradiction? ]]>

x,y,z are reals]]>

1) $6x^2+x(19y+3)+15y^2+5y=0$

2)$10x^3+(11y+5)x^2+(3y^2+3y+12)x+6y+6=0 $]]>

-----

Let $0\le x \le \dfrac{\pi}{2}$. Prove that $\sin x \ge \dfrac{2x}{\pi}$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

posted a screenshot because I think the wording is awkward and I think it is easy to setup it up wrong

so to begin with we have a and b being list of added numbers

$\dfrac{a}{100}=23$ and $\dfrac{a+b}{150}=27$

so now what?]]>

Zp is a vector space if and only if p is prime]]>

For all A,B,C.........we have:

1) A=A

2) A=B <=> B=A

3) A=B & B=C => A=C

4) A=B => A+C= B+C

5) A=B=> AC =BC ( NOTE :Instead of writing A.C or B.C e.t.c we write AB ,BC e.t.c)

6) A+B= B+A.......................................................AB=BA

7) A+(B+C) = (A+B)+C.............................................................A(BC)=(AB)C

10) A+0=A..................................................................................1A=A

11)...

equivalences]]>

1) $x^2(y-w)=a$

2) $y^2(w-x)=b$

2) $w^2(x-y) =c$

a,b,c non zero

2) $y^2(w-x)=b$

2) $w^2(x-y) =c$

a,b,c non zero

Iterated integrals

Q1, 2,3

Double integrals in polar coordinates

Q1, 2,3

Triple integrals

Q1, 2,3

Triple integrals in cylindrical coordinates

Q1, 2,3

Triple integrals in spherical coordinates

Q1, 2,3

Change of variables

Q7,8,9

Green's theorem

Q1,2

Surface integrals

Q1,2,3

Divergence theorem

Q1,2,3

Stokes theorem

Q1,2,3]]>

I am looking the following:

Solve for a fix number $c\in \mathbb{R}$ the following linear system of equations: $$\begin{cases}x_1-cx_2+(2c-1)x_3=-(c+1) \\ 3x_2+(5c+8)x_3=-(c-2)\end{cases}$$

For which values of $c$ is there one solution and for which values are there no solution?

I have done the following:

First we write this in matrix form:

\begin{equation*}\begin{pmatrix}\left.\begin{matrix}1 & -c & 2c-1 \\ 0 & 3 & 5c+8 \end{matrix}\right|\begin{matrix}-(c+1) \\...

For which c is there 1/0 solution?]]>

I have proven the following properties:

(i) $(a\Rightarrow b)\iff (\neg b\Rightarrow \neg a)$

(ii) $[(a\Rightarrow b)\land (b\Rightarrow c)]\Rightarrow (a\Rightarrow c)$

(iii) $[a\land (a\Rightarrow b)]\Rightarrow b$

At the second part of the exercise we have to show the de Morgan rules:

- $C_X(A\cup B)=(C_XA)\cap (C_XB)$

- $C_X(A\cap B)=(C_XA)\cup (C_XB)$

where $A,B$ are subsets of $X$ and $C_X$ is the complement in $X$.

Is the first part relevant, i.e. do we have to...

De Morgan Rules - Logical statements]]>

I am looking at the following exercise but I think that I miss something.

The statement is the following:

We are given the following system of equations: \begin{align*}2a-2c+d-2e=&-2 \\ -2c-2d+2e=&\ \ \ \ \ 3 \\ d+2e=&-2\end{align*}

1) Is the system in echelon form? Justify.

2) Solve the linear system of equations over $\mathbb{R}$.

3) How many solutions has the system?

The system in matrix form is: \begin{equation*}\begin{pmatrix}\left.\begin{matrix}2 & -2 & 1 & -2 \\...

Linear system of equations: Echelon form/Solutions]]>

I want to draw a circuit that has four inputs a, b, c, d, and which gives the value $[(a \lor b) \lor (c \lor \neg d)] \land (c \land \neg a)$ at the output, using exclusively the NAND gates with two inputs.

Hint: If you connect the two inputs of such a NAND gate, you get a so-called inverter, i.e. a circuit implementation of the logical negation.

The NAND-gate is:

I have done the following:

\begin{align*}[(a\lor b)\lor (c\lor \neg d)]\land...

Draw a NAND-gate to get that output]]>

If f and g are injective, then so is gβf | ||

If f and g are surjective, then so is gβf | ||

If f and g are bijective, then so is gβf | ||

If gβf is bijective, then so are both f and g | ||

If gβf is bijective, then so is... |

Composing functions]]>

S is reflexive

S is symmetric

S is transitive

S is reflexive and symmetric

S is reflexive and transitive

I assume that S acts on a set A. So let a,b,c be elements of A.

For S to be reflexive, for all a in A, a S a.

For S to be symmetric, for all a,b in A, if a S b, then b S a.

For S to be transitive, for all a,b,c in A, if a S b and b S c, then a S c.

Now I got to compose S with S, and I know that...

Relation composition]]>

I want to prove by using the rules of boolean algebra that the following statement is always true $$\{b\land [\neg a\Rightarrow \neg b]\} \Rightarrow a$$

Since we have to use the rules of boolean algebra, we cannot use the truth table, right?

Could you give me a hint how we could show that?

]]>

$$g(x) = \frac{A}{\exp(-bQ+c)}\Big(\frac{1 + \exp(-bQ+c)}{1 + \exp(-dx+c)}-1\Big)$$

I suppose I should start by considering an event $\Phi$ such that $\Phi = \mathbb{P}[x \geq Q]$. However, I don't know how to go around this condition.

All constants are positive...

How to calculate conditional expectation E[g(x) | x>= Q] for x ~ exp(1)]]>

Visit my new website where you can try out my math software for free and leave a comment:

https://gillesvanderveken.wixsite.com/t ... ion-solver

Enter the commands 1b, 2b, ... , 20b for demo's.

Once you see how it works, enter the commands 1, 2, ..., 20 using your own input.

Regards,

Gilles Vanderveken]]>

I am looking at the following:

translate the following statements into set inclusion.

(i) Those who drown are not a fish or a swimmer.

(ii) Scientists are human.

(iii) A person who is not a swimmer is a non-swimmer.

(iv) Fish are not human.

(v) There was a case of a drowned mathematician.

(vi) Mathematicians are scientists.

Check if from the statements (i)β(vi)

,,There was a mathematician who was not a swimmerβ

can be implied.

I have done the following:

We consider the sets:

E...

Translate the statements into set inclusion]]>

-----

In a convex quadrilateral $PQRS$, $PQ=RS$, $(\sqrt{3}+1)QR=SP$ and $\angle RSP-\angle SPQ=30^{\circ}$. Prove that $\angle PQR-\angle QRS=90^{\circ}$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

I'm not sure how to get this first derivative (mainly where does the 4 come from?)

I know xΜ is the sample mean (which I think is 1/2?)

Can someone suggest where to start with finding the log-likelihood?

I know the mass function of a binomial distribution is:

Thanks!]]>

-----

Let $a,\,b,\,c$ be three real numbers such that $1\ge a \ge b \ge c \ge 0$. Prove that if $k$ is a (real or complex) root of the cubic equation $x^3+ax^2+bx+c=0$, then $|k|\le 1$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

\begin{equation}

E = \tau\Big[a\frac{1+a}{1+e^{-bx+c}} - 1 \Big]^+

\end{equation}

where $[z]^+ = \max(z,0)$

How can I find

- The PDF for $E$
- The expected value of E.

Here are the questions with unknown answers. (These chemistry questions are other math topics.)

1) At 1 bar, the boiling point of water is 372.78 K. At this temperature and pressure, the density of liquid water is $958.66 kg/m^3$ and that of gaseous water is $0.59021 kg/m^3.$ What are the molar volumes. in $m^3 mol^{-1}$ of liquid and gaseous water at this temperature and pressure? in Liters/mol?

2) Refer to the answer to 1) question. Assuming that a water molecule excludes the...

Thermodynamics questions]]>

ok without any calculation I felt it could not be determined since we have one equation with 2 variables]]>

Find a general solution to the system of differential equations

$\begin{array}{llrr}\displaystyle

\textit{given}

&y'_1=\ \ y_1+2y_2\\

&y'_2=3y_1+2y_2\\

\textit{solving }

&A=\begin{pmatrix}1 &2\\3 &2\end{pmatrix}\\

\textit{eigensystem}.

&\begin{pmatrix}1-\lambda &2\\3 &2-\lambda\end{pmatrix}

=\lambda^2-3\lambda -4 = (\lambda-4)(\lambda+1) = 0 \\

&\lambda = 4,-1

\end{array}$

so...

307.27.1]]>

]]>

Can you please check it for me that I have done it wrong or not ?

Thank you in advance.]]>

x | f(A)=B | |

x | In the arrow diagram representing f, every point in B has an arrow pointing at it. | |

x | βyβB βxβA such that f(x)=y | |

x | fβ1(B)=A | |

x | Every element of... |

Definition of onto function]]>

$\textit{not to scale}$

A summer camp counselor wants to find a length, x,

The lengths represented by AB, EB BD,CD on the sketch were determined to be 1800ft, 1400ft, 7000ft, 800 ft respectfully

Segments $AC$ and $DE$ intersect at $B$, and $\angle AEB$ and $\angle CDE$ have the same measure What is the value of $x$?

looks easy but still tricky

$\dfrac{x}{EB}=\dfrac{CD}{BD}

=\dfrac{x}{1400}=\dfrac{800}{700}$

multiple thru by 1400 then simplify...

gre.ge.2 distance by similiar triangles]]>

$\textbf{xy-plane}$ above shows one of the two points of intersection of the graphs of a linear function and and quadratic function.

The shown point of intersection has coordinates $\textbf{(v,w)}$ If the vertex of the graph of the quadratic function is at $\textbf{(4,19)}$,

what is the value of $\textbf{v}$?

${-6}\quad {6}\quad {5}\quad {7}\quad {8}$

ok before I plow into this one it seems...

-ge.04 GRE intersection of parabola and line]]>

Riders in an amusement park ride shaped like a Viking ship hung from a large pivot are rotated back and forth like a rigid pendulum. Sometime near the middle of the ride, the ship is momentarily motionless at the top of its circular arc. The ship then swings down under the influence of gravity.

(a) Assuming negligible friction, find the speed of the riders at the bottom of its arc, given the systemβs center of mass travels in an arc having a radius of 14.0 m and the riders are near the...

Centripetal acceleration of viking ship in the amusement park.]]>

If this behavior depends on the initial value of y at t = 0,describe the dependency

\begin{array}{lll}

\textit{rewrite}

&y'-2y=-3\\ \\

u(t)

&=\exp\int -2 \, dx=e^{-2t}\\ \\

\textit{product}

&(e^{-2t}y)'=-3e^{-2t}\\ \\

\textit{integrate}

&e^{-2t}y=\int -3e^{-2t} \, dt =\dfrac{3e^{-2t}}{2}+c\\ \\

%e^{-2t}y&=\dfrac{3e^{-2t}}{2}+c\\ \\

\textit{isolate}

&y(t)=\dfrac{3}{2}+\dfrac{c}{e^{-2t}}\\ \\

t \to \infty&=\dfrac{3}{2}+0 =\dfrac{3}{2}...

b.1.1.2 behavior as t goes to infinity]]>

]]>