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# Differential Equations/ODEs

Question
Hi,

I'm currently learning about ordinary first and second order differential equations and while I have no problem arriving at the correct solution, I'm struggling to understand what the solution actually gives me. Whether I'm dealing with ordinary separable equations or ordinary 1st order equations, I end up with an equation for y(x), but what does that equation actually represent.

Calculus can be used in several places.  A paper on it is calculus.nipissingu.ca/calc_app.html

For one example, take the height of stone on earth that is falling.  Let t be a measure of time, where t = 0 is where the problem is started.  If h(t) is the height of a stone, then v(t) is the velocity, and a(t) is the  acceleration.  In mathematical form, we can take y(t) = h(t), and this gives y'(t) = v(t) and y"(t) = a(t).

In this problem, let's take the original height as H0, the original velocity V0,
and the original acceleration A0.  On earth, A0 = 9.81 { in metric, m/sē }.

Since the equation h(t) measures how far up the stone is, gravity makes the acceleration be negative.  That is, a(t) = -9.81.  Note that a(t) is also y"(t), so we have y"(x) = -9.81.

The integral of acceleration a(t) with respect to time t gives the velocity v(t).
That gives v(t) = -9.81t + V0.  As said, this is also y'(t).

The integral of velocity is the height.  Since velocity is v(t) = -9.8t + V0,
that makes the height h(t) be given by h(t) = -4.905tē + V0*t + H0.

Differential Equations

Volunteer

#### Scott A Wilson

##### Expertise

I am capable of solving most ordinary differential equations and a few not so ordinary, but those are few and far between.

##### Experience

I have taken Differential Equations at OSU. Occasionally I have assisted people with questions on the subject.

Publications
I wrote a thesis on the study of shock waves and rarefaction fans. That occurs in nature when mathematics fails to apply. See, mathematics says that when shock waves appear, there should be two solutions whereas nature knows there is only one way to be in that area. When rarefaction fans occur, the mathematics says there should be no solution, but despite this, nature still moves ahead with its own plan.

Education/Credentials
I received an MS degree at OSU in Mathemematics and a BS degree at OSU in Mathematical Science.

Awards and Honors
I graduated from OSU with a Bachelor of Science degree in mathematics with honors. Two years later, I graduated from OSU with a Master of Science degree in mathematics with honors. Between these two degrees, I have more hours in graduate courses that required to get them.

Past/Present Clients
I answered many questions at OSU, down in south seattle at a college, at a church in Corvallis, OR, as Safeway in Washington, and many other areas. I have answered over 8,500 right here, but only a little over 190 of them have been in differential equations.