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We describe numerical and graphical methods for understanding differential
We cannot (yet!) solve the differential equation However, from the equation alone,
we can describe some facts about the solution.
Consider the solution to the differential equation which passes through the point .
Is increasing or decreasing at ?
By definition, a solution to this differential equation which passes through must
have , and
This is positive, so the function is increasing at .
In fact, we can say more. The differential equation tells us the slope of any solution
passing through a given point.
Consider the solution to the differential equation
which passes through the point . What is the slope of the solution at
a solution to this differential equation which passes through must have , and
Hence the slope is at .
The slope of at is .
Given a differential equation, say , we can pick points in the plane and compute what
the slope of a solution at those points will be. Repeating this process, we can
generate a slope field. The slope field for the differential equation looks like this:
Let’s be explicit:
A slope field, also called a direction field, is a graphical aid for
understanding a differential equation, formed by:
Choosing a grid of points.
At each point, computing the slope given by the differential equation, using
the and -values of the point.
At each point, drawing a short line segment with that slope.
Here is the slope field for the differential equation , with a few solutions of the
differential equation also graphed.
Notice that the slope field suggests one solution to this differential equation, which is
a straight line.
Of the solutions shown in the slope field above, what is the formula for the linear
Autonomous differential equations
Consider the following differential equations
The first differential equation, , is rather easy to solve, we simply integrate both sides.
This type of differential equation is called a pure-time differential equation.
Pure-time differential equations express the derivative of the solution explicitly as a
function of an independent variable. We can symbolically describe a pure-time
differential equation as
On the other hand, the second differential equation, does not involve the
independent variable, , at all! Such differential equations are called autonomous
A differential equation that does not involve the independent variable is called an
autonomous differential equation. We can symbolically write to express an
autonomous differential equation.
Finally the third differential equation, , expresses as a function of both and the
independent variable . Such a differential equation is called a nonautonomous
differential equation. We can symbolically describe a nonautonomous differential
Which of the following are autonomous differential equations?
Since autonomous differential equations only depend on the function’s value their
behavior does not depend on the independent variable,
Which of the five slope fields shown are for autonomous differential equations?
If a differential equation is autonomous, then all of the slopes will be the same on
each horizontal line.
Which of the five slope fields shown are for pure-time differential equations?
If a differential equation is pure-time, then all of the slopes will be the same on each
Which of the five slope fields shown are for differential equations that are neither
autonomous nor pure-time?
If a differential equation is neither autonomous nor pure-time, then the slope will
change along horizontal lines and vertical lines.
Consider the differential equation , whose slope field is given below. Which of the
following statements appear to be true?
The solution passing through the origin has .The solution passing through has .The solution passing through is always decreasing.There are two solutions
which are constant functions.Every solution has a vertical tangent line at
Consider the autonomous differential equation: The constant functions and are
solutions to this differential equation. In fact, for any autonomous differential
equation , where is a function of , if for any constant , then will be a constant
solution to the differential equation. These constant solutions are also known as
equilibrium solutions. We can witness these solutions if we inspect the slope field:
Consider the autonomous differential equation . Which of the following are
Consider the slope field of :
Finally, let’s work an example problem:
Consider the differential equation: Find all equilibrium solutions.
First note that this
differential equation is pure-timeautonomousnonautonomous , and hence any solution where , will be an equilibrium solution. Hence we must
solve the equation Write with me to factor hence and are equilibrium solutions to
this differential equation.
In science and mathematics, finding exact solutions to differential equations is not
always possible. We have already seen that slope fields give us a powerful way to
understand the qualitative features of solutions. Sometimes we need a more precise
quantitative understanding, meaning we would like numerical approximations of the
Again, suppose you have set up the following differential equation If we know that
solves this differential equation, and , how might we go about approximating ? One
idea is to repeatedly use linear approximation.
Let us approximate just using the two subintervals Since , we know that by linear
But now, we have so
Plotting our approximation with the actual solution we find:
This approximation could be improved by using more subintervals. We will now
formalize the method of using repeated linear approximation to approximate
solutions to differential equations, and call it Euler’s Method.
Euler’s Method Consider a differential equation on the interval where is
a function of two variables and it is given that . To approximate on the
Set and .
Decide either a step-size or how many subintervals you want to divide
the interval into. Either way: and .
Iteratively define for .
Then , and so .
Let’s try an example.
Use Euler’s method to approximate on where for the differential equation where .
Using the notation from the definition, we have Fill out the following table for
Euler’s method using subintervals. Thus .