Preface

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This is a tutorial made solely for the purpose of education and it was designed for students taking Applied Math 0330. It is primarily for students who have very little experience or have never used Mathematica before and would like to learn more of the basics for this computer algebra system. As a friendly reminder, don't forget to clear variables in use and/or the kernel.

Finally, the commands in this tutorial are all written in bold black font, while Mathematica output is in normal font. This means that you can copy and paste all commands into Mathematica, change the parameters and run them. You, as the user, are free to use the scripts for your needs to learn the Mathematica program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately.

Exact Equations

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Recall the total differential of a function ψ(x,y) of two variables, denoted by dψ, is given by the expression

Let M(x,y) and N(x,y) be two smooth functions having continuous partial derivatives in some domain \( \Omega \subset \mathbb{R}^2 \) without holes. A differential equation, written in differentials

is called exact if and only if or there exists a smooth function \( \psi (x,y) ,\) called the potential function, such that its total differential A potential function is not unique to which arbitrary constant can be added. Once it is known, the general solution is obtained immediately: Since Mathematica uses the command "N" for numerical evaluation, we change notations and use MM(x,y) and NN(x,y) instead of M(x,y) and N(x,y), respectively.

There are two approaches to find a potential function corresponding to an exact equation. The first one is based on integration of

because the total differential of ψ is \( {\text d} \psi = \psi_x {\text d}x + \psi_y {\text d} y \) for all \( x \in \Omega . \) The second method utilizes line integration, which is prefered for solving initial value problems. Indeed, suppose we are given an initial value problem for an exact equation: Let L be arbitrary curve starting at \( (x_0 ,y_0 ) \) and ending at arbitrary point \( (x,y) . \) This curve or line should be without self-intersections and belong to the domain Ω where functions M(x,y) and N(x,y) possess continuous derivatives. Then the potential function can be obtained as In practice, we choose L as a semi-linear line going parallel to axes from initial point \( (x_0 ,y_0 ) \) and finishing at arbitrary point \( (x,y) \in \mathbb{R}^2 . \) In particular, we can integrate first along vertical axis and then horizontally, or we can integrate horizontally and then vertically.

If we integrate along black line (vertically where dx = 0 and then horizontally where dy = 0), we get

Now if we integrate along blue dashed line (horizontally where dy = 0 and then vertically where dx = 0), we get

Example: The equation \( y \,\text{d}x + x \,\text{d}y =0 \) is exact because \( M_y =1 = N_x \) for \( M= y \quad\mbox{and} \quad N= x . \) Suppose that the initial condition \( y(2)=3 \) is given.

We type in Mathematica:

The solution is psi[x,y]==0:

Define the gradient function:

and then check our potential function:

We check the answer with (blue) line integration: Therefore, the solutiuon becomes ψ =0 or \( xy -6 =0 . \)

Example: Consider the differential equation

First, we check whether the given equation is exact.