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myrzilka [38]
3 years ago
9

The slope field for the differential equation dy/dx=(3y)/(xy+5x) will have vertical segments when

Mathematics
1 answer:
Inessa05 [86]3 years ago
3 0

Answer:

E) x=0 or y=-5

Step-by-step explanation:

The slope field will have vertical segments when dy/dx is undefined, that is, dy/dx is a division by 0.

We have that

dy/dx=(3y)/(xy+5x)

So the slope field will have vertical segments when

xy + 5x = 0

x(y + 5) = 0

This is when x = 0 or y = -5.

So the correct answer is:

E) x=0 or y=-5

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sladkih [1.3K]

Hey there!!

Let's take the first number as ' x '

Then, the second number would be ' x + 1 '

Sum of these = 183

Hence,

x + x + 1 = 183

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2x = 182

x = \frac{182}{2}

x = 91

Hence, the small integer is 91

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6 0
3 years ago
Find the perimeter of a rectangle with a width of (x - 3) and a length of 4x ​
Firdavs [7]

Step-by-step explanation:

        4x

   __________

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x-3|                    |     x-3

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2 years ago
PLEASE HELP <br> 100!!! POINTS
faust18 [17]

Answer:

Most Unlikely:  Less than 5

Likely:  2-digit number

Unlikely:  Multiple of 3

Equally Likely:  Odd

Most Likely:  Greater than 10

Step-by-step explanation:

there are 6x5, or 30 outcomes

Less than 5 has 1 outcome - spinner and dice both are one

Odd has 15 outcomes

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6 0
2 years ago
Solve the following differential equation using using characteristic equation using Laplace Transform i. ii y" +y sin 2t, y(0) 2
kifflom [539]

Answer:

The solution of the differential equation is y(t)= - \frac{1}{3} Sin(2t)+2 Cos(t)+\frac{5}{3} Sin(t)

Step-by-step explanation:

The differential equation is given by: y" + y = Sin(2t)

<u>i) Using characteristic equation:</u>

The characteristic equation method assumes that y(t)=e^{rt}, where "r" is a constant.

We find the solution of the homogeneus differential equation:

y" + y = 0

y'=re^{rt}

y"=r^{2}e^{rt}

r^{2}e^{rt}+e^{rt}=0

(r^{2}+1)e^{rt}=0

As e^{rt} could never be zero, the term (r²+1) must be zero:

(r²+1)=0

r=±i

The solution of the homogeneus differential equation is:

y(t)_{h}=c_{1}e^{it}+c_{2}e^{-it}

Using Euler's formula:

y(t)_{h}=c_{1}[Sin(t)+iCos(t)]+c_{2}[Sin(t)-iCos(t)]

y(t)_{h}=(c_{1}+c_{2})Sin(t)+(c_{1}-c_{2})iCos(t)

y(t)_{h}=C_{1}Sin(t)+C_{2}Cos(t)

The particular solution of the differential equation is given by:

y(t)_{p}=ASin(2t)+BCos(2t)

y'(t)_{p}=2ACos(2t)-2BSin(2t)

y''(t)_{p}=-4ASin(2t)-4BCos(2t)

So we use these derivatives in the differential equation:

-4ASin(2t)-4BCos(2t)+ASin(2t)+BCos(2t)=Sin(2t)

-3ASin(2t)-3BCos(2t)=Sin(2t)

As there is not a term for Cos(2t), B is equal to 0.

So the value A=-1/3

The solution is the sum of the particular function and the homogeneous function:

y(t)= - \frac{1}{3} Sin(2t) + C_{1} Sin(t) + C_{2} Cos(t)

Using the initial conditions we can check that C1=5/3 and C2=2

<u>ii) Using Laplace Transform:</u>

To solve the differential equation we use the Laplace transformation in both members:

ℒ[y" + y]=ℒ[Sin(2t)]

ℒ[y"]+ℒ[y]=ℒ[Sin(2t)]  

By using the Table of Laplace Transform we get:

ℒ[y"]=s²·ℒ[y]-s·y(0)-y'(0)=s²·Y(s) -2s-1

ℒ[y]=Y(s)

ℒ[Sin(2t)]=\frac{2}{(s^{2}+4)}

We replace the previous data in the equation:

s²·Y(s) -2s-1+Y(s) =\frac{2}{(s^{2}+4)}

(s²+1)·Y(s)-2s-1=\frac{2}{(s^{2}+4)}

(s²+1)·Y(s)=\frac{2}{(s^{2}+4)}+2s+1=\frac{2+2s(s^{2}+4)+s^{2}+4}{(s^{2}+4)}

Y(s)=\frac{2+2s(s^{2}+4)+s^{2}+4}{(s^{2}+4)(s^{2}+1)}

Y(s)=\frac{2s^{3}+s^{2}+8s+6}{(s^{2}+4)(s^{2}+1)}

Using partial franction method:

\frac{2s^{3}+s^{2}+8s+6}{(s^{2}+4)(s^{2}+1)}=\frac{As+B}{s^{2}+4} +\frac{Cs+D}{s^{2}+1}

2s^{3}+s^{2}+8s+6=(As+B)(s²+1)+(Cs+D)(s²+4)

2s^{3}+s^{2}+8s+6=s³(A+C)+s²(B+D)+s(A+4C)+(B+4D)

We solve the equation system:

A+C=2

B+D=1

A+4C=8

B+4D=6

The solutions are:

A=0 ; B= -2/3 ; C=2 ; D=5/3

So,

Y(s)=\frac{-\frac{2}{3} }{s^{2}+4} +\frac{2s+\frac{5}{3} }{s^{2}+1}

Y(s)=-\frac{1}{3} \frac{2}{s^{2}+4} +2\frac{s }{s^{2}+1}+\frac{5}{3}\frac{1}{s^{2}+1}

By using the inverse of the Laplace transform:

ℒ⁻¹[Y(s)]=ℒ⁻¹[-\frac{1}{3} \frac{2}{s^{2}+4}]-ℒ⁻¹[2\frac{s }{s^{2}+1}]+ℒ⁻¹[\frac{5}{3}\frac{1}{s^{2}+1}]

y(t)= - \frac{1}{3} Sin(2t)+2 Cos(t)+\frac{5}{3} Sin(t)

3 0
3 years ago
Lola is saving money for a new bike. The first month, she saves $50 from her birthday. Lola plans to save $10 per month thereaft
weqwewe [10]
So this can be translated like the following:
Bike costs= 173 dollars She currently has 107. She saves 11 dollar per week to get the bike so the formula is:
173 = 107 + 11X where X is the number of weeks

Step 1: bring the 107 to the other side to get 173-107 = 11X
Step 2: divide both sides by 11 to solve for X to get this (173-107 )/11 = X
Step 3: Calculate and simplify 66 / 11 = X which is the same as X= 6

So the answer is it will take jennifier 6 weeks to save for the bike.
8 0
3 years ago
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