Differential Equations Test

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Differential Equations Test - 10

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Question 1
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
A differential equation of the form where P and Q are functions of x alone, is called a first order linear differential equation. It has many applications in physics including RL circuits.
Consider the linear differential equation
Q. The general solution is _______.

A
A.

B
B.

C
C.

D
D.

Solution

The general solution is
Question 2
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
A differential equation of the form where P and Q are functions of x alone, is called a first order linear differential equation. It has many applications in physics including RL circuits.
Consider the linear differential equation
Q. The value of ∫ P dx = ______.

A
A.

B
B. log x² + C

C
C. log|x| + C

D
D.

Solution

= 2log |x| + C
= log x² + C
Question 3
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
A differential equation of the form where P and Q are functions of x alone, is called a first order linear differential equation. It has many applications in physics including RL circuits.
Consider the linear differential equation
Q. ∫ Qdx = _______.

A
A. log|x| + C

B
B. 2x + C

C
C. log x² + C

D
D.

Solution

The given differential equation can be expressed as
P = 2/x , Q = x
∫ Qdx = ∫ xdx
Question 4
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
A differential equation of the form where P and Q are functions of x alone, is called a first order linear differential equation. It has many applications in physics including RL circuits.
Consider the linear differential equation
Q. The integrating factor is _______.

A
A. log |x|

B
B. 2x

C
C. x²

D
D. e^{log x}

Solution
Question 5
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
A differential equation of the form where P and Q are functions of x alone, is called a first order linear differential equation. It has many applications in physics including RL circuits.
Consider the linear differential equation
Q. If y(1) = 0, then y(2) = _______.

A
A. 0

B
B. 1

C
C. 15/4

D
D. 15/16

Solution

Given y(1) = 0
⇒
⇒ C = -1/4
The particular solution is
When x = 2, 4y = 15/4
⇒ y = 15/16
Question 6
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
Polio drops are delivered to 50K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation dy/dx = k(50 - y) where x denotes the number of weeks and y the number of children who have been given the drops.
Q. The value of C in the particular solution given that y(0) = 0 and k = 0.049 is

A
A. log 50

B
B. log 1/50

C
C. 50

D
D. -50

Solution

Given,
y(0) = and k = 0.049
We have, -log|50 - y| = Kx + C
log |50 - y| = -Kx - C
log |50 - 0| = 0 - C
[∵ x = 0, K = 0.049, y(0) = 0]
log 50 = - C
C = log 1/50
Question 7
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
Polio drops are delivered to 50K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation dy/dx = k(50 - y) where x denotes the number of weeks and y the number of children who have been given the drops.
Q. Which method of solving a differential equation can be used to solve dy/dx = k(50 - y)?

A
A. Variable separable method

B
B. Solving Homogeneous differential equation

C
C. Solving Linear differential equation

D
D. All of the above

Solution

‘Variable separable method’ is used to solve such an equation in which variables
can be separated completely, i.e., terms containing x should remain with dx and terms containing y should remain with dy.
Question 8
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
Polio drops are delivered to 50K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation dy/dx = k(50 - y) where x denotes the number of weeks and y the number of children who have been given the drops.
Q. State the order of the above given differential equation.

A
A. 2

B
B. 1

C
C. 0

D
D. Can’t define
Question 9
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
Polio drops are delivered to 50K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation dy/dx = k(50 - y) where x denotes the number of weeks and y the number of children who have been given the drops.
Q. The solution of the differential equation dy/dx = k(50 - y) is given by,

A
A. log |50 – y| = kx + C

B
B. –log |50 – y| = kx + C

C
C. log |50 – y| = log|kx|+ C

D
D. 50 – y = kx + C

Solution

dy/dx = k(50 - y)
Question 10
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
Polio drops are delivered to 50K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation dy/dx = k(50 - y) where x denotes the number of weeks and y the number of children who have been given the drops.
Q. Which of the following solutions may be used to find the number of children who have been given the polio drops?

A
A. y = 50 – e^kx

B
B. y = 50 – e^–kx

C
C. y = 50(1 – e^–kx )

D
D. y = 50(e^–kx – 1)

Solution

We have