# Practice Questions for Standard Differential Equations (12 Questions)

Please see the attached files for the fully formatted problems.

This question is concerned with finding the solution of the first order simultaneous equations

where a = -2, b = 8, c = -24, d = 30

(i) Find the particular solutions to the differential equations which satisfy the initial conditions

x = 16 and y = 3 at t = 0.

For this part of the question give x as a function of t.

Omit the "x = " (8 marks)

Your Answer: 45/2e^(6t)-13/2e^(22t)

Comment: The particular solution required is x = ""

(ii) What is y as a function of t.

Omit the "y = "

(8 marks)

Your Answer: 45/2e^(6t)-39/2e^(22t)

Comment: The particular solution required is y = ""

(iii) What is the value of x at t = 0.05?

Give your answer to AT LEAST TWO PLACES OF DECIMALS. Put in this value only ie. Omit the "x = " (2 marks)

Your Answer: 10.84474402

Comment: x = 10.84474402

(iv) What is the value of y at t = 0.05?

Give your answer to AT LEAST TWO PLACES OF DECIMALS. Put in this value only ie. Omit the "y = " (2 marks)

Your Answer: -28.20941429

Comment: y = -28.20941429

Question 1: Score 0/0

In a chemical plant, two connected tanks each contain a suspension of solid precipitate. The liquid in the tanks is caused to flow back and forth between the tanks, resulting in varying concentrations of the precipitate in each tank.

The concentrations in tank 1 and tank 2, measured relative to a fixed positive reference concentration, are given by respectively at time . They are governed by the following pair of first order, coupled differential equations:

where a = -5, b = 9, c = -27, d = 31

(i) Find the concentrations of the precipitate in each tank if the initial concentrations in tank 1 and tank 2 are and .

For this part of the question give the concentration in tank 1 as a function of t.

Omit the " = " (8 marks)

Your Answer: 3/2e^(4t)+1/2e^(22t)

Comment: The particular solution required in this part is = ""

(ii) What is as a function of t?

Omit the " = "

(8 marks)

Your Answer: 3/2e^(4t)+3/2e^(22t)

Comment: The particular solution required in this part is = ""

(iii) What is the concentration in tank 1 at t = 0.04?

Give your answer to AT LEAST TWO PLACES OF DECIMALS. Put in this value only

ie. omit the " = " (2 marks)

Your Answer: 2.965716159

Comment: = 2.965716159

(iv) What is the concentration in tank 2 at t = 0.04?

Give your answer to AT LEAST TWO PLACES OF DECIMALS. Put in this value only

ie. omit the " = " (2 marks)

Your Answer: 5.376615865

Comment: = 5.376615865

Two pressure vessels in a plant are connected by a series of pipes and filters. The pressures in each vessel are different in this dynamic situation.

The pressures, relative to a constant reference level, in vessel 1 and vessel 2 are given by respectively at time . They are governed by the following pair of first order, coupled differential equations:

where a = -18, b = 7, c = -14, d = 3

(i) Find the pressures in each pressure vessel if the initial pressures in vessel 1 and vessel 2 are and .

For this part of the question give the pressure in pressure vessel 1 as a function of t.

Omit the " = " (8 marks)

Your Answer: (-5e^(-4t))+15e^(-11t)

Comment: The particular solution is = ""

(ii) What is as a function of t?

Omit the " = "

(8 marks)

Your Answer: (-10e^(-4t))+15e^(-11t)

Comment: The particular solution is = ""

(iii) What is the pressure in pressure vessel 1 at t = 0.06?

Give your answer to AT LEAST TWO PLACES OF DECIMALS. Put in this value only

ie. omit the " = " (2 marks)

Your Answer: 3.819630712

Comment: = 3.819630712

(iv) What is the pressure in tank 2 at t = 0.06?

Give your answer to AT LEAST TWO PLACES OF DECIMALS. Put in this value only

ie. omit the " = " (2 marks)

Your Answer: -.113508593

Comment: = -.113508593

#### Solution Preview

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Q1.

(*)

where a=-2 b=8 c=-24 d=30

(1) Find particular solutions satisfying initial conditions x=16 and y=3 at t=0. For this part of the question give x as a function of t.

Solution. We can write (*) as a form of matrix.

.......(**)

So, we have a matrix

Now we need to find its eigenvalues by solving the following equation

So,

gives two eigenvalues

So, has two eigenvalues .

Now we need to find the eigenvectors

When , we have

So, we can choose a vector

When , we have

So, we can choose a vector

Now we can form a matrix

We can find the inverse of P is

Also, we can check

Now we do linear transformation

where . Then by (**), we have

So,

Note that

We have

i.e.

So, its solution is

So, we get general solution to original variables x and y is

=

.................................(***)

Now we use initial condition ...

#### Solution Summary

12 Practice Questions for Standard Differential Equations are solved. The solution is detailed and well presented.