Question

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

Answer 1

− 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

Explanation:

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

`a_ny^n+a_n_-_1y^(n-1)+...+a_1y'+a_oy=0`

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

`a_nr^n+a_n_-_1r^(n-1)+...+a_1r+a_o=0`

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.

• If the discriminant is greater than zero, the solution is over-damped

• If the discriminant is less than zero, the solution is under-damped

• If the discriminant is equal to zero, the solution is critically damped

So, given the differential equation:

`-3y''-3y+3y=0`

Which has characteristic equation of the form:

`-3r^2-3r+3=0`

The quadratic polynomial of the form:

`ar^2+br+c=0`

Has discriminant:

`Disc=b^2-4ac`

In this case:

`a=-3\\b=-3\\c=3`

So:

`Disc=(-3)^2-4(-3)(3)=9-(-36)=45`

In this case:

`Disc=45>0`

Therefore the solution is over-damped.

Now, given the differential equation:

`-2y''-4y'+1y=0`

Which has characteristic equation of the form:

`-2r^2-4r+1=0`

The quadratic polynomial of the form:

`ar^2+br+c=0`

Has discriminant:

`Disc=b^2-4ac`

In this case:

`a=-2\\b=-4\\c=1`

So:

`Disc=(-4)^2-4(-2)(1)=16+8=24`

In this case:

`Disc=24>0`

Therefore the solution is over-damped.

Finally, given the differential equation:

`1y''+7y'+5y=0`

Which has characteristic equation of the form:

`1r^2+7r+5=0`

The quadratic polynomial of the form:

`ar^2+br+c=0`

Has discriminant:

`Disc=b^2-4ac`

In this case:

`a=1\\b=7\\c=5`

So:

`Disc=(7)^2-4(1)(5)=49-20=29`

In this case:

`Disc=29>0`

Therefore the solution is over-damped.