Elimination of Arbitrary Constants (Differential Equations)

[Portal.Leaf]

Homework Statement

Eliminate the arbitrary constants of the equation:
ax² + bx + c

Homework Equations

(Concept) According to my instructor, having n arbitrary constants makes an nth-order differential equation.

The Attempt at a Solution

I tried to differentiate 'til I get a third derivative so I ended with the final answer y''' = 0. Is my answer correct? If not, I'm willing to learn.

Thank you!

 

[statdad]

Since your post does not contain an equation, and the item you gave does not itself contain any derivatives, what you want isn't clear.

 

[1MileCrash]

What you are doing is writing a differential equation from a solution, essentially. I take it you mean y = ax² + bx + c.

Your solution, y''' = 0, is correct. Here is how you can check this.

If y''' = 0, then by integration, y'' = a.
Then by integration again y' = ax + b.
Then by integration a third time, y = ax² + bx + c, which is the equation you started with.

Basically, elimination of arbitrary constants is a terrible way to say "find a differential equation for which this is the general solution." Therefore you can check your work by solving your resulting differential equation. If the general solution to your resulting differential equation is the equation you started with in the problem, you got it right.

 

[HallsofIvy]

If you are given that $$y= ax^2+ bx+ c$$, what is y'? What is y''? What is y'''? Do you see that the constants have been eliminated?

 

[Portal.Leaf]

Thanks for all your replies. 1MileCrash was right. The answer is indeed y''' = 0. Thanks for helping me check my answer!