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.

figure c

step-by-step explanation:

hope this .

1. x> -5

2. x< 8

step-by-step explanation:

1. 2x> -10

divide by 2 from both sides of equation.

2x/2> -10/2

simplify, to find the answer.

-10/2=-5

x> -5 is the correct answer.

2. 3x< 24

divide by 3 from both sides of equation.

3x/3< 24/3

simplify, to find the answer.

24/3=8

x< 8 is the correct answer.

i hope this you, and have a wonderful day!