? + (350 × 0.84) = (620 × 0.55)

Find the solution to the differential equation dB/dt+4B=20 with B(1)=30

natita [175]

Answer:

The solution of the differential equation is $B=5+25e^{-4t+4}$

Step-by-step explanation:

The differential equation $\frac{dB}{dt}+4B=20$ is a first order separable ordinary differential equation (ODE). We know this because a separable first-order ODE has the form:

$y'(t)=g(t)\cdot h(y)$

where g(t) and h(y) are given functions.

We can rewrite our differential equation in the form of a first-order separable ODE in this way:

$\frac{dB}{dt}+4B=20\\\frac{dB}{dt}=20-4B\\\frac{dB}{dt}=4(5-B)\\\frac{1}{5-B}\frac{dB}{dt}=4$

Integrating both sides

$\frac{1}{5-B}\frac{dB}{dt}=4\\\frac{1}{5-B}\cdot dB=4\cdot dt\\\\\int\limits {\frac{1}{5-B}} \, dB=\int\limits {4} \, dt$

The integral of left-side is:

$\int\limits {\frac{1}{5-B}} \, dB\\\mathrm{Apply\:u-substitution:}\:u=5-B\\\int\limits {\frac{1}{5-B}} \, dB=\int\limits {\frac{1}{u}} \, dB\\\mathrm{du=-dB}\\-\int\limits {\frac{1}{u}} \,du\\\mathrm{Use\:the\:common\:integral}:\quad \int \frac{1}{u}du=\ln \left(\left|u\right|\right)\\-\int\limits {\frac{1}{u}} \,du =-\ln \left|u\right|\\\mathrm{Substitute\:back}\:u=5-B\\-\ln \left|5-B\right|\\\mathrm{Add\:a\:constant\:to\:the\:solution}\\-\ln \left|5-B\right|+C$

The integral of right-side is:

$\int\limits {4} \, dt = 4t + C$

We can join the constants, and this is the implicit general solution

$-\ln \left|5-B\right|+C=4t + C\\-\ln \left|5-B\right|=4t + D$

If we want to find the explicit general solution of the differential equation

We isolate B

$-\ln \left|5-B\right|=4t + D\\\ln \left|5-B\right|=-4t+D\\\left|5-B\right|=e^{-4t+D}$

Recall the definition of |x|

$|x|=\left \{ {{x, \:if \>x\geq \>0 } \atop {-x, \:if \>x0}} \right.$

So

$\left|5-B\right|=e^{-4t+D}\\5-B= \pm \:e^{-4t+D}\\B=5 \pm \:e^{-4t+D}\\B=5\pm \:e^{-4t}\cdot e^{D}\\B=5+Ae^{-4t}$

where $A=\pm e^{D}$

Now B(1) =30 implies

$B=5+Ae^{-4t}\\30=5+Ae^{-4}\\30-5=Ae^{-4}\\25e^{4}=A$

And the solution is

$B=5+(25e^{4})e^{-4t}\\B=5+25e^{-4t+4}$

8 0

4 years ago

The ratio of papayas to mangoes is 3:4 and the ratio of papaya to apple is 5:2. If there are 27 pieces of apple, hoe many mangoe

bija089 [108]

I m not sure of the answer but this is my best try

Papaya: Mango = 3:4
Papaya: Apple = 5:2
Lets take no of papaya as x

So, 5/2 = x/27
By cross- multiplication, x= 27*5/2
x= 67.5

Total number of papayas = 67.5
Lets take number of mangoes as y

So, 3/4 = 67.5/y
By cross multiplication, y= 90

Therefore, total number of mangoes are 90 mangoes.

7 0

3 years ago

Read 2 more answers

The equation e = 1200 + 11t represent your elevation e in feet for each minute t you hike from a trailhead

Lady bird [3.3K]

T=199 maybe wrong or maybe right

4 0

4 years ago

One month, Ruby worked 6 hours more than Isaac, and Svetlana worked 4 times as many hours as Ruby. Together they worked 126 hour

Paladinen [302]

Answer/Step-by-step explanation:

Let x hours be the number of hours Issac worked

x+6 hours be the number of hours Ruby worked

4(x+6) hours be the number of hours Svetlana worked.

x + x + 6 + 4(x+6) = 126

x + x + 6 + 4x +24 = 126

6x +30 = 126

6x = 96

x = 16

Therefore, Issac worked 16 hours.

Ruby worked 16+6 = 22 hours

Svetlana worked 22 × 4 = 88 hours.

If helpful, please mark as brainliest! =)

6 0

3 years ago

Read 2 more answers

In an architecture class the students built a scale model of an office building. If the scale is 1 in. = 12 feet and the model i

iogann1982 [59]

Answer:

102 feet tall

Step-by-step explanation:

12 multiplied by 8 1/2 to get 102

7 0

3 years ago

Read 2 more answers

? + (350 × 0.84) = (620 × 0.55) – √1764 2