Answer:
− 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:
[tex]a_ny^n+a_n_-_1y^{n-1}+...+a_1y'+a_oy=0[/tex]
This differential equation will have a characteristic equation of the form:
[tex]a_nr^n+a_n_-_1r^{n-1}+...+a_1r+a_o=0[/tex]
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:
[tex]-3y''-3y+3y=0[/tex]
Which has characteristic equation of the form:
[tex]-3r^2-3r+3=0[/tex]
The quadratic polynomial of the form:
[tex]ar^2+br+c=0[/tex]
Has discriminant:
[tex]Disc=b^2-4ac[/tex]
In this case:
[tex]a=-3\\b=-3\\c=3[/tex]
So:
[tex]Disc=(-3)^2-4(-3)(3)=9-(-36)=45[/tex]
In this case:
[tex]Disc=45>0[/tex]
Therefore the solution is over-damped.
Now, given the differential equation:
[tex]-2y''-4y'+1y=0[/tex]
Which has characteristic equation of the form:
[tex]-2r^2-4r+1=0[/tex]
The quadratic polynomial of the form:
[tex]ar^2+br+c=0[/tex]
Has discriminant:
[tex]Disc=b^2-4ac[/tex]
In this case:
[tex]a=-2\\b=-4\\c=1[/tex]
So:
[tex]Disc=(-4)^2-4(-2)(1)=16+8=24[/tex]
In this case:
[tex]Disc=24>0[/tex]
Therefore the solution is over-damped.
Finally, given the differential equation:
[tex]1y''+7y'+5y=0[/tex]
Which has characteristic equation of the form:
[tex]1r^2+7r+5=0[/tex]
The quadratic polynomial of the form:
[tex]ar^2+br+c=0[/tex]
Has discriminant:
[tex]Disc=b^2-4ac[/tex]
In this case:
[tex]a=1\\b=7\\c=5[/tex]
So:
[tex]Disc=(7)^2-4(1)(5)=49-20=29[/tex]
In this case:
[tex]Disc=29>0[/tex]
Therefore the solution is over-damped.
