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2 years ago

Solve the differential equation. y' + 5xey = 0.

Mathematics

1 answer:

galben [10]2 years ago

5 0

Answer:

The solution is $y = - ln(\frac{5}{2}x^{2} + C)$

Step-by-step explanation:

To solve the differential equation, we will find $y$

From the given equation, y' + 5xey = 0.

That is, $y' + 5xe^{y} = 0$

This can be written as

$\frac{dy}{dx} + 5xe^{y} = 0$

Then,

$\frac{dy}{dx} = - 5xe^{y}$

$\frac{dy}{e^{y}} = - 5x dx$

Then, we integrate both sides

$\int {\frac{dy}{e^{y}}} =\int {- 5x dx}$

$\int {e^{-y}dy }} =\int {- 5x dx}$

Then,

$-e^{-y} = -\frac{5}{2}x^{2} + C$

$e^{-y} = \frac{5}{2}x^{2} + C$

Then,

$ln(e^{-y}) = ln(\frac{5}{2}x^{2} + C)$

Then,

$-y = ln(\frac{5}{2}x^{2} + C)$

Hence,

$y = - ln(\frac{5}{2}x^{2} + C)$

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2 years ago

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2 years ago

Let p represent any number in the interval (- 18,22). Rewrite the possible values for p in inequality notation.

Nataliya [291]

Answer:

$-18 < p < 22$

Step-by-step explanation:

We are given the following information in the question.

The open interval: (- 18,22).

Let p belong to the given interval then p represents all the values belonging to the given interval.

Open Interval: An open interval is a interval that does not include the end points.

Thus, we can write:

$-18 < p < 22$

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2 years ago

3y''-6y'+6y=e*x sexcx

Simora [160]

From the homogeneous part of the ODE, we can get two fundamental solutions. The characteristic equation is

$3r^2-6r+6=0\iff r^2-2r+2=0$

which has roots at $r=1\pm i$ . This admits the two fundamental solutions

$y_1=e^x\cos x$
$y_2=e^x\sin x$

The particular solution is easiest to obtain via variation of parameters. We're looking for a solution of the form

$y_p=u_1y_1+u_2y_2$

where

$u_1=-\displaystyle\frac13\int\frac{y_2e^x\sec x}{W(y_1,y_2)}\,\mathrm dx$
$u_2=\displaystyle\frac13\int\frac{y_1e^x\sec x}{W(y_1,y_2)}\,\mathrm dx$

and $W(y_1,y_2)$ is the Wronskian of the fundamental solutions. We have

$W(e^x\cos x,e^x\sin x)=\begin{vmatrix}e^x\cos x&e^x\sin x\\e^x(\cos x-\sin x)&e^x(\cos x+\sin x)\end{vmatrix}=e^{2x}$

and so

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Therefore the particular solution is

$y_p=\dfrac13e^x\cos x\ln|\cos x|+\dfrac13xe^x\sin x$

so that the general solution to the ODE is

$y=C_1e^x\cos x+C_2e^x\sin x+\dfrac13e^x\cos x\ln|\cos x|+\dfrac13xe^x\sin x$

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2 years ago

Can someone please help me in one of these!?

Anarel [89]

Answer:

9) y = -200x + 1200

11) slope = -2

15) $2,450

16) 9 days

Step-by-step explanation:

9) Given the two points from the graph:

Let (x₁, y₁) = (0, 1200)

(x₂, y₂) = (1, 1000)

Substitute these values into the following slope formula:

m = (y₂ - y₁)/(x₂ - x₁)

m = (1000 - 1200)/(1 - 0)

m = -200/1

m = -200

The slope of the line is -200.

Next, we need to determine the y-intercept, which is the point on the graph where it crosses the y-axis. Upon observing the graph, it shows that the line crosses at point (0, 1200). The y-coordinate of this ordered pair is the value of the y-intercept, b = 1200.

Therefore, the linear equation in slope-intercept form is y = -200x + 1200.

<h3>11) Given the points, (5, -18) and (-4, 0): </h3>

Let (x₁, y₁) =(5, -18)

(x₂, y₂) = (-4, 0)

Substitute these values into the following <u>slope formula</u>:

m = (y₂ - y₁)/(x₂ - x₁)

$m = \frac{0 - (-18)}{-4 - 5} = \frac{0 + 18}{-9} = -2$

Therefore, the slope of the line is -2.

<h3>15) Solve:</h3>

Given the fixed fee of $200, and the $150 per hour after the initial meeting:

We can represent these in slope-intercept form:

y = 150x + 200

y = total cost for the Attorney's services

x = number of hours worked.

The y-intercept in this given problem is $200, which represents the flat fee charged for the initial meeting. While the slope in this equation is $150, which is the hourly rate that an Attorney charges his clients after the initial meeting.

If an Attorney works for 15 hours, then:

Let x = 15, and substitute its value into the equation to find the total cost:

y = 150x + 200

y = 150(15) + 200

y = 2,250 + 200

y = 2,450

Therefore, a client will pay a total of $2,450 for an Attorney's 15 hours of work.

<h3>16) Solve: </h3>

Given the water level on a Lake of 165 inches after a rainstorm, and the water level's receding rate of 3 inches per day:

We can establish the following linear equation to model this given problem:

L = -3d + 165

Where:

L = represents the water level of the Lake

d = number of days that the water level recedes

In order to find the number of days it will take before the water level recedes to 84 inches:

Substitute the value of L = 84 into the equation:

L = -3d + 165

84 = -3d + 165

Subtract 165 from both sides:

84 - 165 = 3d + 165 - 165

-27 = -3d

Divide both sides by -3:

$\frac{-27}{-3} = \frac{-3d}{-3}$

9 = d

Therefore, it will take 9 days for the water level to be at 84 inches.

7 0

2 years ago