Usual convention for proving differential equations

Click For Summary

Discussion Overview

The discussion revolves around the conventions and methods used for proving solutions to ordinary differential equations (ODEs). Participants explore different approaches to validating solutions, contrasting derivation from first principles with verification of given solutions.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant expresses confusion over the method of proving differential equations, advocating for deriving solutions from first principles rather than merely verifying given solutions.
  • Another participant suggests that while deriving solutions is important, it is also standard practice to check a proposed solution by substituting it back into the original differential equation.
  • A later reply indicates that for exam purposes, simply confirming that the left-hand side of the equation corresponds to the right-hand side with a particular solution is sufficient, referring to this as a form of confirmation rather than proof.
  • One participant emphasizes that if a problem states to show that a specific function is a solution, it suffices to substitute the function into the equation and check if it satisfies the conditions, without needing to solve the equation itself.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the preferred method for proving differential equations. There are differing views on whether deriving solutions from first principles or verifying given solutions is more appropriate.

Contextual Notes

Participants mention the context of high school math education and exam requirements, which may influence their perspectives on proof methods.

Schreiberdk
Messages
93
Reaction score
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?
 
Physics news on Phys.org
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?
 
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.
 
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
 
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?
 

Similar threads

Undergrad Why Lagrange’s method of solving Pp + Qq=R works?
  • · Replies 3 ·
Replies
3
Views
2K
Insights Differential Equation Systems and Nature
  • · Replies 14 ·
Replies
14
Views
6K
High School Real world applications of differential equations
  • · Replies 5 ·
Replies
5
Views
6K
Graduate Solution to a differential equation with variable coefficients
  • · Replies 6 ·
Replies
6
Views
2K
High School Basic Idea of Differential Equations
  • · Replies 25 ·
Replies
25
Views
4K
Undergrad Proving a basis is a basis for homogeneous linear differential equation with constant (complex) coefficients
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Second order non-homogeneous linear ordinary differential equation
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Correct Upper/Lower limits for continuation of solutions for ODE
  • · Replies 0 ·
Replies
0
Views
4K
High School Incorporating boundary conditions in the Finite Element Method (FEM)
  • · Replies 4 ·
Replies
4
Views
4K
Undergrad On error estimates of approximate solutions
  • · Replies 1 ·
Replies
1
Views
3K