Which linear inequality is represented by the graph?

A function y(t) satisfies the differential equation dy dt = y 4 − 6y 3 + 5y 2 . (a) What are the constant solutions of the equat

Alexxandr [17]

Answer:

Hence increasing in (-\infty,0) U (1,5)

c) Decreasing in (0,1)

Step-by-step explanation:

Given that y(t) satisfies the differential equation

$\frac{dy}{dt} =y^4-6y^2+5y^2\\=y^2(y^2-6y+5)\\=y^2(y-1)(y-5)$

Separate the variables to have

$\frac{dy}{y^2(y-1)(y-5)} =dt$

Left side we can resolve into partial fractions

Let $\frac{1}{y^2(y-1)(y-5)} =\frac{A}{y} +\frac{B}{y^2}+\frac{C}{y-1} \frac{D}{y-5}$

Taking LCD we get

$1= Ay(y-1)(Y-5) +B(y-1)(y-5)+Cy^2 (y-5)+Dy^2 (y-1)\\Put y =1\\1 = -4C\\Put y =5\\ 1 = 25(4)D\\Put y =0\\1=5B\\$

By equating coeff of y^3 we have

A+C+D=0

$C=\frac{-1}{4} \\D=\frac{1}{100} \\B =\frac{1}{5} \\A = -C-D = \frac{6}{25}$

Hence left side =

$\frac{6}{25y} +\frac{1}{5y^2}+\frac{-1}{4(y-1)}+ \frac{1}{100(y-5)}=dt\\\frac{6}{25}ln y -\frac{1}{5y}-\frac{1}{4}ln|(y-1)| +\frac{1}{100}ln|y-5| = t+C$

b) y is increasing whenever dy/dt>0

dy/dt =0 at points y =0, 1 and 5

dy/dt >0 in (-\infty,0) U (1,5)

Hence increasing in (-\infty,0) U (1,5)

c) Decreasing in (0,1)

5 0

3 years ago

Read 2 more answers

Sam has saved 200 sand dollars and wants to give them away equally to n friends. Write an expression to show how many sand dolla

jeka57 [31]

200 ÷ n = x (or something similar)
To now how many sand dollars each friend will receive we need to now how many friends there are so we can divide the sand dollars up!

5 0

2 years ago

Read 2 more answers

Sharon has some one-dollar bills and some five-dollar bills. she has 14 bills. the value of the bills is $30.

romanna [79]

Four 5 dollar bills and ten 1 dollar bills

5 0

3 years ago

Compute the average speed in m/s of a runner who completes a mile in 4.00 min. Show work please and much appreciated. You will e

LenaWriter [7]

How many seconds are in 4 min 24000
so divide 4
its becomes 6000 thats 1 min.
so divide by 60
it becomes 100
so he has an average speed of 1m/100seconds i think i got lost there

8 0

3 years ago

Which of these lines is perpendicular to the line y = 3x + 2?

Morgarella [4.7K]

Answer:

(c) $y=\frac{-1}{3}x+4$ will be perpendicular to the line y = 3x+2

Step-by-step explanation:

We have given the the equation of line as y = 3x+2

We know the equation of line as y = mx+c , here m is slope and c is intercept

On comparing with standard equation of line m = 3

We know that slope of perpendicular line is $=\frac{-1}{m}$

So the slope of the line which is perpendicular to y = 3x+2 will be $=\frac{-1}{3}$

Now (a) Line is given as $y=\frac{1}{3}x+3$ its slope is $\frac{1}{3}$ so it will be not perpendicular

(b) Line is given as $y=\frac{1}{3}x+2$ its slope is $\frac{1}{3}$ so it will be not perpendicular

(c) Line is given as $y=\frac{-1}{3}x+4$ its slope is $\frac{-1}{3}$ so it will be perpendicular to the line y = 3x+2

(d) Line is given as $y=3x+\frac{1}{2}$ its slope is 3 so it will not perpendicular to the line y = 3x+2

3 0

2 years ago