Differential equation tutorial pdf


















One of the most useful features of a symbolic manipulation system is the ability to simplify mathematical expressions. SymPy has dozens of functions to perform various kinds of simplification. There is also one general function called simplify that attempts to apply all of these functions in an intelligent way to arrive at the simplest form of an expression. Here are some examples. We see that simplify is capable of handling a large class of expressions.

But simplify has a pitfall. It just applies all the major simplification operations in SymPy, and uses heuristics to determine the simplest result. We did not get what we want. There is a function to perform this simplification, called factor , which will be discussed below.

Another pitfall to simplify is that it can be unnecessarily slow, since it tries many kinds of simplifications before picking the best one. If you already know exactly what kind of simplification you are after, it is better to apply the specific simplification function s that apply those simplifications.

Applying specific simplification functions instead of simplify also has the advantage that specific functions have certain guarantees about the form of their output. These will be discussed with each function below. For example, factor , when called on a polynomial with rational coefficients, is guaranteed to factor the polynomial into irreducible factors. It is entirely heuristical, and, as we saw above, it may even miss a possible type of simplification that SymPy is capable of doing.

You may then choose to apply specific functions once you see what simplify returns, to get a more precise result. It is also useful when you have no idea what form an expression will take, and you need a catchall function to simplify it. Although it has a lot of scopes, for now, we will consider its function in expanding polynomial expressions. Given a polynomial, expand will put it into a canonical form of a sum of monomials. After all, by its very name, it makes expressions bigger, not smaller.

Usually this is the case, but often an expression will become smaller upon calling expand on it due to cancellation. For polynomials, factor is the opposite of expand. Note that the input to factor and expand need not be polynomials in the strict sense. They will intelligently factor or expand any kind of expression though note that the factors may not be irreducible if the input is no longer a polynomial over the rationals.

Note that since factor will completely factorize both the numerator and the denominator of an expression, it can also be used to do the same thing:. However, if you are only interested in making sure that the expression is in canceled form, cancel is more efficient than factor. For example, the inverse cosine, or arc cosine, is called acos.

To simplify expressions using trigonometric identities, use trigsimp. Before we introduce the power simplification functions, a mathematical discussion on the identities held by powers is in order. There are three kinds of identities satisfied by exponents. Identity 2 is not always true. Identity 3 is not always true.

This is important to remember, because by default, SymPy will not perform simplifications if they are not true in general. In order to make SymPy perform simplifications involving identities that are only true under certain assumptions, we need to put assumptions on our Symbols. We will undertake a full discussion of the assumptions system later, but for now, all we need to know are the following.

That is, a simplification will not be applied to an expression with a given Symbol unless it holds for all complex numbers. Symbols can be given different assumptions by passing the assumption to symbols.

For the rest of this section, we will be assuming that x and y are positive, and that a and b are real. We will leave z , t , and c as arbitrary complex Symbols to demonstrate what happens in that case.

They are exactly the same object. Notice that powsimp refuses to do the simplification if it is not valid. This will force the simplification to take place, regardless of assumptions. Note that in some instances, in particular, when the exponents are integers or rational numbers, and identity 2 holds, it will be applied automatically.

This means that it will be impossible to undo this identity with powsimp , because even if powsimp were to put the bases together, they would be automatically split apart again. As with powsimp , identity 2 is not applied if it is not valid.

In SymPy, as in Python and most programming languages, log is the natural logarithm, also known as ln. As before, z and t will be Symbols with no additional assumptions. As always, the identities will not be applied unless they are valid.

To apply identities 1 and 2 from right to left, use logcombine. SymPy implements dozens of special functions, ranging from functions in combinatorics to mathematical physics.

An extensive list of the special functions included with SymPy and their documentation is at the Functions Module page. We will also define k , m , and n. The factorial function is factorial. The binomial coefficient function is binomial. The factorial function is closely related to the gamma function , gamma. The generalized hypergeometric function is hyper.

A common way to deal with special functions is to rewrite them in terms of one another. This works for any function in SymPy, not just special functions. To rewrite an expression in terms of a function, use expr. For example,. For some tips on applying more targeted rewriting, see the Advanced Expression Manipulation section.

To rewrite hyper in terms of more standard functions, use hyperexpand. To simplify combinatorial expressions, use combsimp. A continued fraction is an expression of the form. A continued fraction can also be infinite, but infinite objects are more difficult to represent in computers, so we will only examine the finite case here.

The easiest way to construct a continued fraction from a list is to work backwards. The symbols function that we have been using has a shortcut to create numbered symbols. This form is useful for understanding continued fractions, but lets put it into standard rational function form using cancel.

Now suppose we were given frac in the above canceled form. In fact, we might be given the fraction in any form, but we can always put it into the above canonical form with cancel. Suppose that we knew that it could be rewritten as a continued fraction. How could we do this with SymPy? This means we need to use the apart function. You can execute multiple lines at once in SymPy Live.

Typing Shift-Enter instead of Enter will enter a newline instead of executing. So try the following exercise. Take a list of symbols and randomize them, and create the canceled continued fraction, and see if you can reproduce the original list. This section covers how to do basic calculus tasks such as derivatives, integrals, limits, and series expansions in SymPy. If you are not familiar with the math of any part of this section, you may safely skip it.

To take derivatives, use the diff function. To take multiple derivatives, pass the variable as many times as you wish to differentiate, or pass a number after the variable. You can also take derivatives with respect to many variables at once. Just pass each derivative in order, using the same syntax as for single variable derivatives.

The two ways of calling diff are exactly the same, and are provided only for convenience. To create an unevaluated derivative, use the Derivative class. It has the same syntax as diff. To evaluate an unevaluated derivative, use the doit method. These unevaluated objects are useful for delaying the evaluation of the derivative, or for printing purposes.

They are also used when SymPy does not know how to compute the derivative of an expression for example, if it contains an undefined function, which are described in the Solving Differential Equations section.

To compute an integral, use the integrate function. There are two kinds of integrals, definite and indefinite. To compute an indefinite integral, that is, an antiderivative, or primitive, just pass the variable after the expression. Note that SymPy does not include the constant of integration.

If you want it, you can add one yourself, or rephrase your problem as a differential equation and use dsolve to solve it, which does add the constant see Solving Differential Equations. For example, to compute. As with indefinite integrals, you can pass multiple limit tuples to perform a multiple integral. If integrate is unable to compute an integral, it returns an unevaluated Integral object. As with Derivative , you can create an unevaluated integral using Integral.

To later evaluate this integral, call doit. Here is a sampling of some of the power of integrate. SymPy can compute symbolic limits with the limit function. The syntax to compute. Like Derivative and Integral , limit has an unevaluated counterpart, Limit. To evaluate it, use doit. SymPy can compute asymptotic series expansions of functions around a point. Order terms can be created and manipulated outside of series. They automatically absorb higher order terms. If you do not want the order term, use the removeO method.

The O notation supports arbitrary limit points other than 0 :. So far we have looked at expressions with analytical derivatives and primitive functions respectively. One approach would be to use a finite difference approach. We can use arbitrary steps possibly containing symbolic expressions :. First order recurrences Second order recurrences Generating functions Series solutions for the first order equations Series solutions for the second order equations Generalized series solutions.

Contents [hide]. NameError : name 'x' is not defined. Quick Tip SymPy expressions are immutable. No function will change them in-place. Warning sympify uses eval. My sine. Note that this is only accurate for small x. Lagrange's method Method of undetermined coefficients Operator methods not sure yet Applications 1. Select a textbook to see worked-out Solutions. Books by Martin Braun with Solutions. Braun, M Braun, … archived file. We do not want to include such discus Differential equations and their applications solution manual 1.

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