Question 1

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically.    &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

11eaa595_6529_3168_a696_13f42e6ca35d_TB4584_11_TB4584_11

Question 2

Solve the homogeneous differential equation   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

11eaa595_6527_d1ca_a696_5917363b0f51_TB4584_11_TB4584_11

Question 3

A calf that weighs 75 pounds at birth gains weight at the rate where w is weight in pounds and t is time in years. If the animal is sold when its weight reaches 750 pounds, find the time of sale using the model . Round your answer to two decimal places.

A) 8.17 years
B) 2.88 years
C) 0.92 year
D) 1 year
E) 0.32 year

1 year

Accepted Answer

1 year

Question 4

Find the general solution of the differential equation   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The general solution of the given differential equation can be found by separating variables:
dy/dx = y^2 - 4y + 3
dy/(y^2-4y+3) = dx
Now, we use partial fraction decomposition to integrate the left-hand side:
dy/[(y-1)(y-3)] = dx/(y-1) - dx/(y-3)
Integrating both sides, we get:
-ln|y-1| + ln|y-3| = x + C
Simplifying and exponentiating both sides, we get:
y = (3-e^(-x+C))/(1-e^(-x+C))
Which can be further simplified as:
y = (3-A)/B
where A and B are constants.
Therefore, the general solution of the given differential equation is y = (3-A)/B , which is closest to option B.

Question 5

A calf that weighs 70 pounds at birth gains weight at the rate   where w is weight in pounds and t is time in years. Use a computer algebra system to solve the differential equation for  &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The given differential equation is:
dW/dt = 11W^(0.595)
Separating variables and integrating we get:
∫ 1/W^(0.595) dW = ∫ 11 dt
Integrating both sides we get:
1/(0.405W^(0.405)) = 11t + C
where C is the constant of integration.
Solving for W we get:
W = [1/(11t + C)]^(2.471)
Since the calf weighs 70 pounds at birth, we can find the value of C:
70 = [1/(0 + C)]^(2.471)
C = -0.104
Substituting this value of C, we get:
W = [1/(11t - 0.104)]^(2.471)
Simplifying this expression we get:
W = 11eaa595_6529_cdad_a696_ebe7b2e18ed6_TB4584_11_TB4584_11

Question 6

Find the particular solution of the differential equation   that satisfies the initial condition   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Find the particular solution of the differential...

Question 7

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically.    &#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

From the slope field, we can see that the solution curves have a negative slope at positive y-values and a positive slope at negative y-values. This suggests that the general solution will involve a natural logarithm.

Integrating the differential equation gives:

$$\int \frac{dy}{y\sqrt{1-y^2}} = \int dx$$

Using the substitution $u = 1-y^2$, we get:

$$-\frac{1}{2}\int \frac{du}{\sqrt{u}} = \int dx$$

$$-\sqrt{u} = 2x + C$$

Substituting back, we get:

$$-\sqrt{1-y^2} = 2x + C$$

Squaring both sides and solving for y gives:

$$y = \pm\sqrt{1-(2x+C)^2}$$

This is the general solution, and we can see that it matches the slope field in choice C.

Question 8

Find the orthogonal trajectories of the family   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Find the orthogonal trajectories of the family...

Question 9

Find an equation of the graph that passes through the point (7, 3) and has the slope   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The point-slope form of a linear equation is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line.
We are given the point (7, 3) and the slope 11eaa595_6526_c14e_a696_7dd83da8be30_TB4584_11_TB4584_11. 
Substituting these values, we get y - 3 = 11eaa595_6526_c14e_a696_7dd83da8be30_TB4584_11_TB4584_11(x - 7).
Simplifying the equation, we get y = 11eaa595_6526_e761_a696_ab734bee0580_TB4584_11_TB4584_11x - 11eaa595_6526_c14e_a696_7dd83da8be30_TB4584_11_TB4584_11*7 + 3. 
Therefore, the answer is C.

Question 10

Determine whether the function   is homogeneous and determine its degree if it is.&#10;A) homogeneous, the degree is 4&#10;B) homogeneous, the degree is 5&#10;C) not homogeneous&#10;D) homogeneous, the degree is 3&#10;E) homogeneous, the degree is 2

Accepted Answer

A homogeneous function is a function in which all its terms have the same degree. However, in the given function, there are two different terms: 11eaa595_6527_3584_a696_013c9b49dc6c_TB4584_11 and TB4584_11. As they have different variables and coefficients, they cannot have the same degree. Hence, the given function is not homogeneous.

Question 11

Find the orthogonal trajectories of the family   .&#10;A)  &#10;B)  &#10;C)  &#10;D)  &#10;E)

Accepted Answer

The answer of Find the orthogonal trajectories of the family...

Question 12

A calf that weighs 80 pounds at birth gains weight at the rate   where w is weight in pounds and t is time in years. What is the maximum weight of the animal if one uses the model   ?&#10;A) 1500 lb&#10;B) 880 lb&#10;C) 1420 lb&#10;D) 1580 lb&#10;E) 2300 lb

Accepted Answer

The answer of A calf that weighs 80 pounds at...

Deck 6: Section 3: Differential Equations