How can you tell if the solution to a differential equation is zero?

Question

In our differential equations class, we are supposed to always "guess" (identically) zero as a solution to a differential equation. However, he doesn't do a good job explaining why that is, or how to check to see if zero is a solution.

Could ya help?

Answer 1

It's usually because when you're solving a differential equation, say in terms of dy/dx, it's typical that y=0 is something that fits the equation. For example, if you have dy/dx = y and want to solve for y, then y=0 actually fits the equation. After all, if you take the derivative of zero you just get zero back, which is also equal here to y. But that's the "trivial solution" because it's so trivial, or "No duh". The other solution that we're much more interested in is y=e^x, which solves the differential equation in a non-trivial way.



In other situations you might know you have a certain number of solutions total, and want to find the trivial solution to get it out of the way so that you can find the "real" solution you're looking for.



EDIT: I wrote all of this before you added the additonal info, but hopefully this should still answer your question.

Answer 2

A differential equation is an equation involving a function and its derivatives.



So to answer your question I guess you would take your equation, calcuate each of its dirvatives (as it appears in the differential equation) plug in the values you are solving for (I assume you are solving it at a specific x,y point) a see if it equals zero.

Answer 3

If y=0 then

y'=0

y''=0

y'''=0

etc



So lets say you have y''+3y'+y=0. Plug in 0 for y'',y',and y and you will get 0=0 and thus y=0 is a solution. But if you had y''+3y'+y=27 and you plug in y=0 then you will get 0=27 which is false and y=0 is not a solution.

Answer 4

Let y = 0; naturally each of y', y'', etc. will be 0 as well. Use this info to see if the resulting equation is a true statement.

Answer 5

I guess its trivial solution.