# Transient-Response Analysis of 1st Order Systems

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CST 8233 – Assignment #3
Transient-Response Analysis of 1st Order Systems
Purpose: Find the solution of 1
st order Ordinary Differential Equations (ODE) using well known
methods; namely, Euler’s and Runge-Kutta 4
nd Order Methods.
Discussion: For the thin, glass-walled mercury thermometer system shown in Figure 1, assume that
the temperature of the bath changes based on certain chemical process occurring between two
substances reacting with each other inside the bath. It is found that the equation that describes this
process is given as follows:
𝑑𝜃(𝑡)
𝑑𝑡 + 2𝜃(𝑡) = cos 4𝑡
It can be found that the actual solution of the response of the thermometer, 𝜃(𝑡), is given by the
following equation:
𝜃(𝑡) = 0.1 cos 4𝑡 + 0.2 sin 4𝑡 + 2.9 𝑒
−2𝑡
Figure 1 Thin, glass-walled mercury thermometer system
The ODE given above can be solved using many numerical methods, such as Euler’s and RungeKutta 4
nd Order Methods.
1. Write a C/C++ program that computes the solution 𝜃(𝑡) using Euler’s Method. For this step, use
the following information: ℎ = 0.8, 0.2, 0.05, 𝜃0 = 3 ℃, 0 ≤ 𝑡 ≤ 2 second. Find the discrete values
of 𝜃(𝑡) at each ℎ step value.
2. Modify the previous code and implement Runge-Kutta 4
th method to solve the same ODE using
the following information: ℎ = 0.8, 0.2, 0.05, 𝜃0 = 3 ℃, 0 ≤ 𝑡 ≤ 2 second. Find the discrete values
of 𝜃(𝑡) at each ℎ step value.
3. Calculate the relative error of the resultant solution at each time for each ℎ step. Your output of
your code should show a table that shows the exact temperature, the estimated temperature, and
the relative error. The user will choose one method and one step size.
What to Submit: Use Brightspace to submit this assignment as a zip file (not RAR, not 9zip, not 7
zip) containing the source code file (ass3.cpp) and a pdf file that shows the plots (bonus). The name
of the zipped folder must contain your name as a prefix so that I can identify it, for example using my
name the file would be kadriaAss3CST8233.zip. It is also vital that you include the file header (as
specified in the Submission Standard) so the file can be identified as yours. Use comment lines in the
There is a late penalty of 25% per day – even one minute is counted late.
You may lose 60% or more if:
⚫ The output is wrong
⚫ Your application won’t build in Visual Studio 2019
⚫ Your application crashes in normal operation
⚫ I can’t build it because you submitted the wrong files or the files are missing, even if it’s an
honest mistake – this gets 100% deduction.
Don’t send me the file as an email attachment – it will get 0.
Example Output
The output of the code should look like below. The results of test case when using Euler’s and RungeKutta for h = 0.2 are shown in the table below.
>> Choose the method for solving the ODE:
1. Euler’s Method
2. Runge-Kutta 4th Order Method
>> 1
>> Choose step size “h” (0.8, 0.2, 0.05)
>> 0.2
Time(second) Exact Temp(C) Estimated Temp(C) Percentage Error(%)
0.2 2.157 2.000 7.28
0.4 1.500 1.339 10.71
0.6 0.935 0.798 14.66
0.8 0.474 0.331 30.13
1.0 0.176 -0.001 100.54
1.2 0.073 -0.131 280.86
1.4 0.128 -0.061 148.01
1.6 0.241 0.118 50.86
1.8 0.299 0.270 9.76
2.0 0.236 0.283 19.89
>> Choose the method for solving the ODE:
1. Euler’s Method
2. Runge-Kutta 4th Order Method
>> 2
>> Choose step size “h” (0.8, 0.2, 0.05)
>> 0.2
Time(second) Exact Temp(C) Estimated Temp(C) Percentage Error(%)
0.2 2.157 2.157 0.01
0.4 1.500 1.500 0.02
0.6 0.935 0.935 0.03
0.8 0.474 0.474 0.06
1.0 0.176 0.176 0.15
1.2 0.073 0.073 0.34
1.4 0.128 0.128 0.15
1.6 0.241 0.241 0.05
1.8 0.299 0.299 0.02
2.0 0.236 0.236 0.00
>> Choose the method for solving the ODE:
1. Euler’s Method
2. Runge-Kutta 4th Order Method

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