Notes to a video lecture on http://www.unizor.com
Higher Order Ordinary
Differential Equations -
Acceleration
Differential equations can include derivatives of higher order - second derivative, third, etc.
Probably, most common equations of this type are those with the second order derivative.
These equations are very often occur in science, especially in Physics. Let's address these equations and approaches to solve them.
Our first subject is a concept of acceleration and Newton's Second Law.
Recall that speed measures how fast a distance from some starting point changes, that is, if this distance is represented as a function x(t) of time t, speedv(t) at any moment t is the first derivative of distance by time:
v(t) = x'(t) = dx/dt
But speed does not have to be constant, we can move faster, increasing our speed (accelerating), or slower, decreasing it (decelerating).
To measure how fast our speed changes with time, as usually, when we want to measure how fast anything changes with time, we use a derivative.
Differentiating speed (a function of time) by time we obtain this measure of change of speed at any moment. This derivative of speed by time is called acceleration a(t):
a(t) = v'(t) = dv/dt =
= x''(t) = d²x/dx²
That is, acceleration is the second derivative of distance by time.
Newton's Second Law states that the force F applied to an object and the acceleration a this object obtains as a result of this application of force are related as follows:
F = m·a
where m is the object's mass (presumed constant).
Assuming that our motion occurs along a straight line with coordinates and, therefore, the position of an object is defined by its X-coordinate x(t), Newton's Second Law is an ordinary differential equation of second order because acceleration is the second derivative of the X-coordinate of an object:
F(t) = m·x''(t)
Usually our task is to find where exactly our object is located (that is, its X-coordinate), if the force, as a function of time, is given.
Consider a case when there is no force applied to an object, that is F(t)=0.
Then, according to the Newton's Second Law,
0 = m·a(t)
from which we derive a(t)=0
Since a(t)=v'(t), we can find the speed:
v'(t)=0
⇒ v(t) = C
(where C is an unknown constant)
⇒ x'(t) = C
⇒ x(t) = C·t + D
(where D is another unknown constant)
That concludes the solution of our differential equation of the second order, and the solution includes two unknown constants that cannot be determined from the equation alone. It's understandable since we don't know initial position of an object on the coordinate axis x(0) and the initial speed it moved v(0). These two additional pieces of information (initial conditions) are needed to determine unknown constants participating in the solution.
If x(0)=x0 and v(0)=v0, we can easily determine
x0 = D and
v0 = C
which results in the final equation of motion of an object, to which no forces are applied (or, more generally, all forces applied to it are balancing each other).
x(t) = x0 + v0·t
By solving the above differential equation of the second order, we have mathematically derived the Newton's First Law as a consequence of the Second Law.
Newton's First Law (law of inertia) states that if the sum of all forces applied to an object is zero, then the object at rest will continue to stay at rest (its speed is and will be 0) and objects moving at some speed will continue to move with the same speed and direction (its speed is constant).
Now consider a case when the force applied to an object is not zero, but constant, that is
F(t)=P=m·a(t)=m·v'(t)
where P is a known constant.
This implies that acceleration a(t) must be a known constant and equals to P/m. Let's use symbol a instead of a(t) to signify this.
Since a=v'(t), we derive
v(t) = a·t + C
where C is an unknown constant.
Then
x'(t) = a·t + C
⇒ x(t) = a·t²/2 + C·t + D
where D is another unknown constant.
To determine two unknown constants we need additional information - initial conditions.
Assume that the original position of an object is x(0)=x0. This allows to determine D=x0.
If initial speed v(0)=v0 is known, we can determine C=v0.
So, the final equation of the motion, when a constant force is applied is
x(t) = a·t²/2 + v0·t + x0
In general, if the force is variable and/or the mass is variable, from Newton's Second Law we can construct a differential equation of the second order, where the second derivative is explicitly represented by a known function:
F(t) = m(t)·x''(t)
⇒ x''(t) = F(t)/m(t)
⇒ d/dt[x'(t)] = F(t)/m(t)
⇒ x'(t) = ∫[F(t)/m(t)]dt
⇒ x(t) = ∫{∫[F(t)/m(t)]dt}dt
As we see, Newton's Second Law presents the simplest kind of ordinary differential equation of the second order. It can be solved by double integration.
It should not be forgotten that in the process of each integration there will appear an unknown constant, to get its value an initial condition should be known and applied.
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