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2 years ago

5

What is the equation in point slope form of a line that passes through the point (–8, 2) and has a slope of 1/2?

1 answer:

Kamila [148]2 years ago

6 0

Point slope equation:
y-y1=m(x-x1)
m=slope

So we simply plug in our given information:
x1=-8
y1=2
m=1/2

y-2=1/2(x-(-8)

2 minus signs next to each other make a positive

Final answer:
y-2=1/2(x+8)

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

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Step-by-step 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:

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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.

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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:

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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:

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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:

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Has discriminant:

$Disc=b^2-4ac$

In this case:

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

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