Use separation of variables to find, if possible, product solutions for the given partial differential equation. (Use the separation constant −λ ≠ 0. If not possible, enter IMPOSSIBLE.)

k∂^2u/∂x^2 - u = ∂u/∂t, k>0

Answers

Answer 1

Answer:

Following are the solution to this question:

Step-by-step explanation:

differentail equation:

[tex]yu_{xy} + u = 0..........(1)[/tex]  

when [tex]u(x,y) = X(x)Y(y)[/tex]

[tex]u_x = \frac{\partial u}{\partial  x} X'Y' \\\\ u_{xy} = \frac{\partial^2 u }{ \partial x\ \partial  y}= X'Y'[/tex]

by substituting the value we get:

[tex]\to \int \frac{Y'}{Y} = -\lambda \int \frac{1}{y} dy\\\\[/tex]

by solving the value:

[tex]\to Y=c_3 y^{-\lambda}\\\\\therefore\ \ c_4=c_2C_3\\\\\to u(x,y) = c_4 e^{\frac{x}{\lambda}} y^{-\lambda}[/tex]

Answer 2

The solution of the given partial differential equation will be

[tex]u(x,y)=c_4e^\frac{x}{\lambda}y^{-\lambda}[/tex]

What is partial differential equation?

The partial derivative is a way to find the slope in either the x or y direction, at the point indicated.

[tex]yu_{xy}+u=0..............................(1)[/tex]

Now if [tex]u(x,y)=X(x)Y(y)[/tex]

[tex]u_x=\dfrac{\partial u}{\partial x} X'Y'[/tex]

[tex]u_{xy}=\dfrac{\partial^2u}{\partial x\ \partial y}=X'Y'[/tex]

by substituting the value, we get:

[tex]\int \dfrac{Y'}{Y}=-\lambda\int\dfrac{1}{y}dy[/tex]

by solving the value:

[tex]Y=c_3y^{-\lambda}[/tex]

[tex]c_4=c_2c_3[/tex]

[tex]u(x,y)=c_4e^\frac{x}{\lambda}y^{-\lambda}[/tex]

Hence the  solution of the given partial differential equation will be

[tex]u(x,y)=c_4e^\frac{x}{\lambda}y^{-\lambda}[/tex]

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Related Questions

WHICH IS EQUIVALENT TO A divided by 2
2A
1/2 A
2/A
A/2

Answers

A/2
1/2 A

Hope this helps!

At the end of a snow storm, Riley saw there was a lot of snow on her front lawn. The temperature increased and the snow began to melt at a steady rate. The depth of snow on Riley's lawn, in inches, can be modeled by the equation S = 13 – t, where t is the time, in hours, after the snow stopped falling. What is the x-intercept of the equation and what is its interpretation in the context of the problem?​

Answers

Answer:

The x-intercept of the function is 13 which represents the number of hours until all the snow has melted.

Step-by-step explanation:

Since there were 0 inches of snow on the lawn after 13 hours, the x-intercept of 13 represents the number of hours after the storm stopped when all of the snow had melted away.

Have a good day :)

The x-intercept of the function is 13 which represents the number of hours until all the snow has melted.

What is coordinate geometry?

A coordinate plane is a 2D plane that is formed by the intersection of two perpendicular lines known as the x-axis and y-axis. A coordinate system in geometry is a method for determining the positions of the points by using one or more numbers or coordinates.

Since there were 0 inches of snow on the lawn after 13 hours, the x-intercept of 13 represents the number of hours after the storm stopped when all of the snow had melted away.

Hence, the x-intercept is 13.

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How many gallons of 40% antifreeze should be mixed with 10 gallons of 80% antifreeze to obtain a 65% antifreeze mixture?

Answers

Equation:

alcohol + alcohol = alcohol

0.40x + 0.85*10 = 0.65(x+10)

----

40x + 85*10 = 65x + 65*10

20*10 = 15x

-----

x = (2/3)20

---

x = 13 1/3 gallons (amt. of 40% solution needed)

Factor this expression completely.
6x^2- 13x - 28

Answers

Answer:

this is the solution

(2−7)(3+4)

Step-by-step explanation:

Answer:

(X+1)(6x-21)

Step-by-step explanation:

6x^2- 13x - 28

6x^2-(21-6)x-28

6x^2-21x+6x-28

X(6x-21)+1(6x-21)

(X+1)(6x-21)

What is the value of x?
7.5/x= 30/16​

Answers

Answer:

4

Step-by-step explanation:

I divided 30/16 and got 1.875, then I knew that 7.5 would have to have the same answer. I did a process of elimination by dividing 7.5 by numbers at random and finally divided by 4 and got the answer. Another way to get the answer would be to cross multiply the equation.

x=4

7.5/4=30/16
120=30x
4=x

What is the equation of the line in slope-intercept form? Enter your answer in the boxes y = __x + __​

Answers

the answer is y=-5x+3 because the -5 is on the x axis and the 3 is on the y axis.

Answer:

y = 3/5x + 3

Step-by-step explanation:

CONGRUENT TRIANGLES WORKSHEET. NEED ASAP DUE TODAY

Answers

Answer:

Congurent triangles are those which map onto them selves perfectly. This means they share the same angle measurements and side lengths. Therefor we can figure out the side lengths by using the information in both triangles.

AD=5x+11

in the congurent triangle, the corresponding side equals:

CM=7x-3

CM and AD are congurent triangles meaning their sidelengths equal the same. Therefore...

5x+11=7x-3

Now we just solve to find the length of AD and CM (equal lengths)

5x+11=7x-3

11=2x-3

14=2x

7=x

x=7

If x=7 then we can plug it in for AD to find the side length.

5(7)+11=46

Notice that CM equals the same 7(7)-3=46

This is because... they are congruent triangles.

These rules also apply for all angle measurements of congruent triangles:

The m<v is equal to the meaasure of E (corresponding angles in congrurent triangles are equal)

We already astablished what x was (x=7) meaning if we find the measure of E then we have the measure of V sicne they are corresponding angles in a congrurent triangle.

(6x7+1)=43°

therefore the m<v is equal to 43°

46 and 43° is your answer

Step-by-step explanation:

the cement truck can pour 20 pounds of cement I 45 seconds how long will it take the cement truck to pour 150 pounds of cement

Answers

337.5 seconds for 150 pounds of cement to be poured

Put these rational numbers in ascending order: 1/2, 55%, 1/3, 0.52
O 1/2, 55%, 1/3, 0.52
O 55%, 0.52, 1/2, 1/3
O 1/3, 1/2, 0.52, 55%
O 1/3, 0.52, 1/2, 55%

Answers

1/3, 1/2, .52, 55%
I believe that’s the correct answer

Answer:

1/3, 1/2, .52, 55%

Step-by-step explanation:

what is (8 + 56) ÷ 4 + 3 2?

Answers

um is the end a 3 and 2 or 32 or 3*2?

Answer:

the answer to your question is 48

What is the slope of the line that contains these points?

Answers

The slope is -4
The change in y over the change in x is -8/2 from x= 1 to x= 3 and from x= 3 to x= 5. -8/2 simplifies to -4

Express 600 as a percentage of 125​

Answers

Answer:

480%

Step-by-step explanation:

600/125*100=480

Use the diagram to solve. What is 260% of 6? Percent Total 100% 100% 20% 20% 20% 260% 3.6 9.6 15.6 21.6

Answers

Answer:

To find this, multiply 6 by the decimal value of the percent.

6*2.6=15.6

Final answer: 15.6

Step-by-step explanation:

Answer:

15.6 or c

Step-by-step explanation:

Please help me I don’t get this Write the equation of the line that is parallel to y=3x-2 and goes through the points (-4,2)

Answers

Alright so answer is y=3x+14

Since you need to find a PARALLEL line to y=3x-2, you already know that the incline is 3x because otherwise the two lines would intersect somewhere.

So for your new equation you now have y=3x+b, so you only need to find b. To do so you’ll need to replace all other variables by numbers with the point that is given and then isolate b.

You can do so with the point that is given to you (-4, 2). You replace y by 2 since the point is (x,y) and replace x by -4.

You now have
2=3(-4)+b
2=-12+b
2+12=b
14=b

You now have your equation: y=3x+14

roger is at the bakery. he is thinking of buying 1 ¾ lb of rainbow cookies. rainbow cookies cost $11.50 per pound. how much would the cookies cost?

Answers

Answer:

The cookies would cost $20.125

Step-by-step explanation:

You would mulitply 1 3/4 by 11.50 to get your answer

Answer:

Step-by-step explanation:

11.50 a pound and he buys 1 3/4 a pound.

11.50*1.75= $20.125 or $20.13 if you round.

A picture is 3 in. wide by 5 in. high was enlarged so that the width was 15 inches. How high is the enlarged picture?

Answers

Answer:

25

Step-by-step explanation:

trust me

Who can access your credit report?
a friends
b. employers
c. neighbors
d. teachers
Please select the best answer from the choices provided.
Ο Α
С В
ОС
D

Answers

This answer would be B

Answer:

B) employers

Step-by-step explanation:

EDGE 2021

Louise Bourgeois obtained a $3,200 installment loan for a marble floor in her foyer. The
annual percentage rate is 9%. She must repay the loan in 24 months. Find the (a) monthly
payment using the formula, (b) total amount repaid, and (c) the finance charge.

Answers

(a) 157.33 (b)3776.00 (c) 576.00

7474 divide 6 answer pleasee

Answers

Answer:

7474/6=1245 2/3

plz help and thankkkkm you​

Answers

Answer:6(x+7= 42

Step-by-step explanation:

A rectangle has a perimeter of 32 in. Find the length and width of the rectangle under which the area is the largest.

a. Let the width to be x and the length to be y , then the quantity to be maximized is (expressed as a function of both x and y) A=_____
b. The condition that x and y must satisfy is y=______

Answers

Answer:

Length of the rectangle is 8 in and the breadth is 8 in.

a. [tex]A=xy[/tex]

b. [tex]y=16-x[/tex]

Step-by-step explanation:

Length = x

Breadth = y

Area is given by [tex]A=xy[/tex]

Perimeter of rectangle is given by

[tex]2(x+y)=32\\\Rightarrow x+y=16\\\Rightarrow y=16-x[/tex]

The condition that x and must satisfy is [tex]y=16-x[/tex]

So, area is

[tex]A(x)=x(16-x)\\\Rightarrow A(x)=16x-x^2[/tex]

Differentiating with respect to x we get

[tex]A'(x)=16-2x[/tex]

Equating with zero

[tex]0=16-2x\\\Rightarrow x=\dfrac{16}{2}\\\Rightarrow x=8[/tex]

Double derivative of A(x)

[tex]A''(x)=-2[/tex]

So [tex]A''(x)<0[/tex] which means [tex]A(x)[/tex] is maximum at [tex]x = 8[/tex]

[tex]y=16-x=16-8\\\Rightarrow y=8[/tex]

So length of the rectangle is 8 in and the breadth is 8 in.

Evaluate 24 + |-27-y| (-7)-x

Answers

Answer:

kr3kf,,xdk,xe3,ke,kex,ks,l,sl,r4kf

Step-by-step explanation:

 nyjogtqGTKJMIYTRFEWDGTYU87R4EW2ERT5YUIO0IUJHYGFREDRGHJKL.BBRTBTRN

Consider the function. f(x) = 2x(x - 1)
What are the values of
f(0)=
f(3)=
f(-4)=

Answers

Answer:

3

Step-by-step explanation:

LET ME KNOW IF I GOT THEM ALL RIGHT PLS

Answers

Answer:

I think so

Step-by-step explanation:

It’s 0 because they dove down and got it

is × = 5 a solution to the equation 19 - × = 14​

Answers

Answer:

19 - 5 = 14

Step-by-step explanation:

this answer is correct

X intercept for -3x+10

Answers

Answer:

(10/3, 0)

Step-by-step explanation:

(10/3, 0)



how this helps

which of the ordered pairs is a solution to the system of inequalities?

Answers

Y>5 which of the orders pairs is a solution to the system of inequality

Mrs. Frazier had 95 inches of ribbon. She gave 14.4 inches to Sara and 27.8 inches to Mary. If she divides the remainder between Jenny and Susie, how much will each girl receive? pls help

Answers

Answer:

26.4

Step-by-step explanation:

95-14.4-27.8=52.8

52.8/2=26.4

Hope I helped!

Please mark Brainliest!!!

evaluate a+b for a= -9 and b=6

Answers

Answer:

-9+6=-3

Step-by-step explanation:

Fill in the variables with their coordinating values. So ...


-9+6=-3


A hot air balloon is at 14 feet above sea level, if you remove 7 sandbags, where
would the hot air balloon be? ( 1 feet = 1 sandbag)

Answers

Answer:

21 feet above sea level.

Step-by-step explanation:

If you remove weight, a hot balloon goes up. 14+7(feet) = 21 feet above sea level.

Answer:

21ft

Step-by-step explanation:

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