All of these ODEs model a system with a spring, mass and dashpot.

Classify the solution for the differential equation: − 3 y ' ' − 3 y ' + 3 y = 0 under-damped over-damped critically damped

Classify the solution for the differential equation: − 2 y ' ' − 4 y ' + 1 y = 0 critically damped under-damped over-damped

Classify the solution for the differential equation: 1 y ' ' + 7 y ' + 5 y = 0 under-damped critically damped over-damped

− 3 y ' ' − 3 y ' + 3 y = 0 : over-damped

− 2 y ' ' − 4 y ' + 1 y = 0 : over-damped

1 y ' ' + 7 y ' + 5 y = 0: over-damped

Step-by-step explanation:

Using the characteristic equation you can express a differential equation of order n as an algebraic equation of degree n:

This differential equation will have a characteristic equation of the form:

Now, you can classify the solution for a differential equation using a simple method. In order to do it, you just need to use the discriminant.

So, given the differential equation:

Which has characteristic equation of the form:

The quadratic polynomial of the form:

Has discriminant:

In this case:

So:

In this case:

Therefore the solution is over-damped.

Now, given the differential equation:

Which has characteristic equation of the form:

The quadratic polynomial of the form:

Has discriminant:

In this case:

So:

In this case:

Therefore the solution is over-damped.

Finally, given the differential equation:

Which has characteristic equation of the form:

The quadratic polynomial of the form:

Has discriminant:

In this case:

So:

In this case:

Therefore the solution is over-damped.

the functions have the same rate of change i think. i narrowed it down to this answer and "the rate of change of tour 1 is greater" but i got it wrong using that answer so im assuming its the answer above. (this is a question i also answered from the iready site if anyone needs a reference)