At the conclusion of this activity, the student will able to

• Understand the concept of exact differential equations.
• to solve the exact differential equations
• plot the graphical representation of solutions of exact differential equations with different initial conditions.

The student will gain the following MATLAB skills:

• How to check if a differential equation is exact or not
• Able to display the result whether the given differential equation is exact or not
• Plot solutions of differential equations for different initial or boundary conditions  using fimplicit 
•  By looking at the plot of the solutions with different conditions, students should be able to recognize solutions to Differential Equations are Functions or Families of Functions.
• Using this plot, the students will be able to identify the real physical situation.

In this lesson, students use MATLAB to find the solution of Exact differential equations and interact with solutions with different initial conditions. This activity is developed as a new initiative to introduce MATLAB for Engineering mathematics students who wish to use MATLAB to solve differential equations.

Prior to working on this activity, students complete an introduction to MATLAB and learn the following: layout of MATLAB, accessing MATLAB documentation, using MATLAB drive, setting directories and paths, performing arithmetic in the command line, creating arrays and variables, using a Livescript, and basic plotting in their MATLAB module. Students also introduced different types of differential equations and the meaning of solutions of differential equations. They are also able to solve an exact differential equation analytically.
They then immediately start working on this activity.

A word document is uploaded .

Make sure the students

• Acquainted with the concept of total differentials 
• are familiar with  basic first order differential equations 
• introduce Exact differential equation in form  M(x, y)dx + N(x, y) dy=0  Explain it is an exact differential form in a domain D in the xy plane if there exists a function F(x, y), called the potential function, such that dF(x, y) = M(x, y)dx + N(x, y)dy for all (x, y) in D.  If   ,  are continuous functions of the differential equation is said to be exact if =
• Checking the condition  to ensure if the differential equation is exact or not. 
• Use Matlab code to check exactness and to solve differential equation with different initial conditions.
• Plots of the potential function F(x,y) to help students visualize level curves.
• Solve problems together as a class, and encourage students to identify whether equations are exact and find the potential function.
• Use real-world problems that model the practical applications of exact equations.

ASSIGNMENT

Uploaded it as a Word document

Resources

Students will be provided with

(1) Notes on Analytic solution of exact differential equations

(2) A live script will be given out to demonstrate how to plot the solution of an exact differential equation. Students are encouraged to use different Matlab plot ideas to plot the solutions of the differential equations.

References

Text Book:  

1. Kreyszig, Erwin. (2011). Advanced engineering mathematics. New York :Wiley .
2. Ross, Shepley L. (1989) Introduction to ordinary differential equations.(4th edition)
3. Mark A. McKibben and Micah D. Webster,((2015). Differential equations with matlab®: exploration, applications, and theory.
4. MATLAB Onramp  -https://www.mathworks.com/learn/tutorials/matlab-onramp.html.