Usual convention for proving differential equations

[Schreiberdk]

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?

 

[Studiot]

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?

 

[Schreiberdk]

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.

 

[Studiot]

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

 

[HallsofIvy]

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 3x^4- 2x^2+ 7x- 54= 0", what would you do?