Usual convention for proving differential equations

In summary, the conversation discusses different methods of proving differential equations and the importance of using known solutions in the process. There is a debate over whether it is necessary to derive solutions from first principles or if it is sufficient to simply check if the solution satisfies the equation. The general consensus is that it is important to check the solution, but not necessarily to derive it every time. The conversation also mentions the use of standard solutions and the importance of substituting the solution into the original equation as a check.
  • #1
Schreiberdk
93
0
Hi there PF

How does mathematicians usually prove differential equation (I am just speaking of the ordinary differential equations).

We are going through proofs of differential equations in my high school math class at the moment, and in our books there are usual proofs, where one derives the solution to the diff. equation from the equation itself, but our teacher (who is a chemical engineer, who has been working a lot with diff. equations) says that it is better to proof them with just taking the solution, equating the lefthand side of the diff. equation and then equating the righthand side of the equation and then showing that they are equivalent.

Im confused, because I think it is better to derive a solution from first principles rather than just showing that it is a right solution.

What do you think and what is the usual convention?
 
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  • #2
When you studied trigonometry, differentiation and integration you were probably shown the principles and the derivations of a few simple examples, cases or 'rules' from first principles.
After that you (hopefully) practised some manipulation so that new problems could be reduced to the table of 'known results' you had.

The solution if differential equation is no exception. There is a table of standard solutions we do not need to derive every time we invoke a DE.

It is however good practice to substitute you solution into the original DE as a check on you working. This is probably what your teacher was doing?
 
  • #3
Yes your probably right :)

It is for our final exam where we get a rule, which we have to proof, and our teacher just says it is a proof to just check if the lefthand side corresponds to the righthand side with the particular solution.
 
  • #4
Schreiberdk said:
Yes your probably right :)

It is for our final exam where we get a rule, which we have to proof, and our teacher just says it is a proof to just check if the lefthand side corresponds to the righthand side with the particular solution.

I would call it confirmation, not proof.

sorry about my earlier poor spelling.

go well
 
  • #5
If the problem says "show that y= f(x) is a solution to the the differentia equation, ..., with initial (or boundary) values..." then the simplest thing to do is just put the function into the equation and initial values and show that it does, in fact satisfy them. If you are given the function, it is NOT necessary to actually solve the equation.

If you were given the problem, "show that x= 2 is a solution to [itex]3x^4- 2x^2+ 7x- 54= 0[/itex]", what would you do?
 

Related to Usual convention for proving differential equations

1. What is the usual convention for proving differential equations?

The usual convention for proving differential equations is to start with the differential equation in its most general form and then use mathematical manipulations and techniques to simplify it and arrive at a solution that satisfies the initial conditions.

2. What are the steps involved in proving differential equations?

The steps involved in proving differential equations usually include identifying the type of differential equation, finding a general solution through integration or other methods, applying initial conditions to find the particular solution, and finally verifying the solution by substituting it back into the original differential equation.

3. How do mathematicians approach the proof of differential equations?

Mathematicians approach the proof of differential equations by using a combination of analytical and numerical methods. They first analyze the equation to determine its type and then use various techniques such as separation of variables, substitution, and series solutions to solve it. They also use numerical methods like Euler's method or Runge-Kutta methods to approximate solutions for more complex equations.

4. Are there any common mistakes to avoid when proving differential equations?

Yes, there are some common mistakes to avoid when proving differential equations. These include errors in algebraic manipulations, missing or incorrect initial conditions, and mistakes in solving the differential equation itself. It is important to double-check all steps and verify the solution by substituting it back into the original equation.

5. Can differential equations be proved without using calculus?

No, differential equations cannot be proved without using calculus. Differential equations are mathematical equations that involve derivatives, and calculus is the branch of mathematics that deals with derivatives and integrals. Without calculus, it would not be possible to manipulate and solve these equations to arrive at a solution.

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