A. Consider the cube shown below. Identify the two-dimensional shape of the cross-section if the cube is sliced horizontally.

Use undetermined coefficient to determine the solution of:y"-3y'+2y=2x+ex+2xex+4e3x

Kitty [74]

First check the characteristic solution: the characteristic equation for this DE is

r ² - 3r + 2 = (r - 2) (r - 1) = 0

with roots r = 2 and r = 1, so the characteristic solution is

y (char.) = C₁ exp(2x) + C₂ exp(x)

For the ansatz particular solution, we might first try

y (part.) = (ax + b) + (cx + d) exp(x) + e exp(3x)

where ax + b corresponds to the 2x term on the right side, (cx + d) exp(x) corresponds to (1 + 2x) exp(x), and e exp(3x) corresponds to 4 exp(3x).

However, exp(x) is already accounted for in the characteristic solution, we multiply the second group by x :

y (part.) = (ax + b) + (cx ² + dx) exp(x) + e exp(3x)

Now take the derivatives of y (part.), substitute them into the DE, and solve for the coefficients.

y' (part.) = a + (2cx + d) exp(x) + (cx ² + dx) exp(x) + 3e exp(3x)

… = a + (cx ² + (2c + d)x + d) exp(x) + 3e exp(3x)

y'' (part.) = (2cx + 2c + d) exp(x) + (cx ² + (2c + d)x + d) exp(x) + 9e exp(3x)

… = (cx ² + (4c + d)x + 2c + 2d) exp(x) + 9e exp(3x)

Substituting every relevant expression and simplifying reduces the equation to

(cx ² + (4c + d)x + 2c + 2d) exp(x) + 9e exp(3x)

… - 3 [a + (cx ² + (2c + d)x + d) exp(x) + 3e exp(3x)]

… +2 [(ax + b) + (cx ² + dx) exp(x) + e exp(3x)]

= 2x + (1 + 2x) exp(x) + 4 exp(3x)

… … …

2ax - 3a + 2b + (-2cx + 2c - d) exp(x) + 2e exp(3x)

= 2x + (1 + 2x) exp(x) + 4 exp(3x)

Then, equating coefficients of corresponding terms on both sides, we have the system of equations,

x : 2a = 2

1 : -3a + 2b = 0

exp(x) : 2c - d = 1

x exp(x) : -2c = 2

exp(3x) : 2e = 4

Solving the system gives

a = 1, b = 3/2, c = -1, d = -3, e = 2

Then the general solution to the DE is

y(x) = C₁ exp(2x) + C₂ exp(x) + x + 3/2 - (x ² + 3x) exp(x) + 2 exp(3x)

4 0

3 years ago

you and 3 friends go to lunch and ordered nachosthat cost 10.30. each person also orders a soda at 1.50. write an equation to sh

kicyunya [14]

Answer:

y = 4(10.30+1.50)

Step-by-step explanation:

Since there are 4 people, you and 3 others, you would multiply everything by 4. Inside the parenthesis, would be the nachos and a drink that everyone got. Since all four people got the same thing, it would be 4x10.30 + 4x1.50, which together would be y = 4(10.30+1.50)

5 0

3 years ago

What shape is produced when slicing a right rectangular pyramid perpendicular to the base

vovikov84 [41]

For the most part, the cross-section formed is a trapezoid, but if the slice passes through the apex of the pyramid, that shape is a triangle.

(Image source: MathCaptain.com)

5 0

3 years ago

What are questions 10-15?

AfilCa [17]

Answer:

10. 105

11. 75

12. 75

13. 105

14. 75

15. 105

Step-by-step explanation:

Notice that 75? Anything that is adjacent (has a common vertex like 75º) to the vertex is a linear pair. here is an example below:

Since there is only THAT 75, you must find the supplementary angle (adds up to 180). and remember in order to get a supplementary angle it has to be a STRAIGHT LINE and HAS TO EQUAL 180. (4, 1 in your image)

180 - 75 = 105.

and also it's also a linear pair so it reflects too which makes things a lot easier!

So, if 4 is 75 and it reflects, then that means 2 is also 75. 1 is 105 and it reflects so 3 has to be 105. and the same for the rest of the numbers.

Hope this makes sense

~R3VO

6 0

3 years ago

Question 1 Which choice shows a 90° clockwise rotation of the figure about the point P?

Arada [10]

The third one is correct.....so go for it without hesitation

6 0

3 years ago