Answer:
(a) The particular solution, y_p is 7
(b) y_p is -4x
(c) y_p is -4x + 7
(d) y_p is 8x + (7/2)
Step-by-step explanation:
To find a particular solution to a differential equation by inspection - is to assume a trial function that looks like the nonhomogeneous part of the differential equation.
(a) Given y'' + 2y = 14.
Because the nonhomogeneus part of the differential equation, 14 is a constant, our trial function will be a constant too.
Let A be our trial function:
We need our trial differential equation y''_p + 2y_p = 14
Now, we differentiate y_p = A twice, to obtain y'_p and y''_p that will be substituted into the differential equation.
y'_p = 0
y''_p = 0
Substitution into the trial differential equation, we have.
0 + 2A = 14
A = 6/2 = 7
Therefore, the particular solution, y_p = A is 7
(b) y'' + 2y = −8x
Let y_p = Ax + B
y'_p = A
y''_p = 0
0 + 2(Ax + B) = -8x
2Ax + 2B = -8x
By inspection,
2B = 0 => B = 0
2A = -8 => A = -8/2 = -4
The particular solution y_p = Ax + B
is -4x
(c) y'' + 2y = −8x + 14
Let y_p = Ax + B
y'_p = A
y''_p = 0
0 + 2(Ax + B) = -8x + 14
2Ax + 2B = -8x + 14
By inspection,
2B = 14 => B = 14/2 = 7
2A = -8 => A = -8/2 = -4
The particular solution y_p = Ax + B
is -4x + 7
(d) Find a particular solution of y'' + 2y = 16x + 7
Let y_p = Ax + B
y'_p = A
y''_p = 0
0 + 2(Ax + B) = 16x + 7
2Ax + 2B = 16x + 7
By inspection,
2B = 7 => B = 7/2
2A = 16 => A = 16/2 = 8
The particular solution y_p = Ax + B
is 8x + (7/2)
Answer:
1561.6
Step-by-step explanation:
I answered this using trial and error, so let me know if you need workings, but I got the dimensions as 18 inches and 13 inches.
I hope this helps!
The height of the <em>water</em> depth is h = 14 + 6 · sin (π · t/6 + π/2), where t is in hours, and the height of the Ferris wheel is h = 21 + 19 · sin (π · t/20 - π/2), where t is in seconds. Please see the image to see the figures.
<h3>How to derive equations for periodical changes in time</h3>
According to the two cases described in the statement, we have clear example of <em>sinusoidal</em> model for the height as a function of time. In this case, we can make use of the following equation:
h = a + A · sin (2π · t/T + B) (1)
Where:
- a - Initial position, in meters.
- A - Amplitude, in meters.
- t - Time, in hours or seconds.
- T - Period, in hours or seconds.
- B - Phase, in radians.
Now we proceed to derive the equations for each case:
Water depth (u = 20 m, l = 8 m, a = 14 m, T = 12 h):
A = (20 m - 8 m)/2
A = 6 m
a = 14 m
Phase
20 = 14 + 6 · sin B
6 = 6 · sin B
sin B = 1
B = π/2
h = 14 + 6 · sin (π · t/6 + π/2), where t is in hours.
Ferris wheel (u = 40 m, l = 2 m, a = 21 m, T = 40 s):
A = (40 m - 2 m)/2
A = 19 m
a = 21 m
Phase
2 = 21 + 19 · sin B
- 19 = 19 · sin B
sin B = - 1
B = - π/2
h = 21 + 19 · sin (π · t/20 - π/2), where t is in seconds.
Lastly, we proceed to graph each case in the figures attached below.
To learn more on sinusoidal models: brainly.com/question/12060967
#SPJ1
Answer:
(x + 9)
Step-by-step explanation:
<u />
<u>X² + 6 - 27</u>
(x-3)
1. Factor the numerator:
<u>(x-3) (x + 9)</u>
(x-3)
2. cancel the (x - 3) from numerator and denominator
3 Answer left is (x + 9)
