In this section we will introduce another important tool for understanding differential equations in the context of an initial value problem. In an initial value problem we are given a differential equation together with an initial condition. Our problem is to describe the particular solution function determined by the given information. Unfortunately, it is rarely possible to find an explicit description of the solution function except in the easiest cases. In most of the harder cases the problem becomes that of finding a good estimate for the value of the solution function. One approach, which we will not pursue here, is to make a sketch of the solution using tangent field and integral curve tools (from section IV.D) and then use that sketch to estimate the desired function value. Instead, we will extend the concept of the linear (or differential) estimate to a more general tool.The basis remains knowing the value of the derivative at a particular point.
Review of The Differential: Recall that knowledge of the value of a function and its derivative at a single point can help us approximate the value of the function at a second point. The technique for this approximation was discussed in Chapter III in the section on the differential. There we developed the formula
We also gave geometric, dynamic, and economic interpretations to this approximation. You might review that section quickly now, since we will use those ideas again to find numerical estimates for the solution of a (first order) differential equation with initial condition. This new method based on the differential estimate is called "Euler's method."
Prelude: The following example may seem simple, but if you understand its reasoning you should have little difficulty with its generalization to Euler's method.
EXAMPLE IV.E.1: Suppose f '(x) = x^{ 2} and f (1) = 3. Estimate f (2) using the differential. Use f '(2) and your estimate of f (2) to estimate f (3).
Solution: Since 2 = 1 + 1 we can use the differential with a = 1 and h = dx = 1. We use the differential equation f '(x) = x^{ 2} to find that f '(1) = 1^{ 2} = 1.
Thus f (2) = f (1+1) ≈f (1) + f '(1)^{ . }1 = 3 + (1)1 = 4.
Now 3 = 2 +1, so we continue to estimate again, this time finding f '(2) = 2^{ 2} = 4
f (3) = f (2 + 1) ≈ f (2) + f '(2)^{ . }1= f (2) + (4) 1 = f (2) + 4.
It is now time to use the previous estimate of f(2) since we have
no better idea of what f(2) might be. Thus f (3) ≈ 4 + 4 = 8.
Interpretations: [graphical] The graphical idea of this technique is illustrated in Figure 1. The line with slope 1 through the point (1,3) gives us the estimate for f (2). Now at the point (2,4) we use the line with slope 4 to obtain the estimate for f(3), at the point (3,8). You can think of this process as going for a walk in the field of tangents determined by this differential equation. We start the trip at the point (1,3) and proceed along the line with slope determined by the differential equation until we come to the point (2,4). Now we use the differential equation at (2,4) to determine the slope of the next line and proceed on that line 'til we come to the point (3,8) where we end our short trip. Instead of walking on the solution curve, we have been travelling for short segments on lines that might be tangents to a solution. Note that we could readjust our lines more frequently thereby taking shorter trips on each segment before reevaluating the slope. Then we would find our path was closer to the solution's graph.
[Motion] The dynamic interpretation of our example is similar. See Figure.2. At time 1 second an object is 3 meters from a point P and its distance from P is increasing at a rate of 1(m/sec). We estimate its position at time 2 to be 4 due to the estimated change in position of 1 meter for the one second of time that has elapsed. Now we find at time 2 seconds that its rate is 4 (m/sec), so we estimate its position at time 3 seconds to be increased 4 meters for the next second of time, giving an estimate for the position at 3 seconds of being 8 meters from P.
[Cost] The economic interpretation of our example is similar. See Figure 2 again. When producing one unit of a product we'll suppose the cost is 3 dollars and its cost is increasing at a rate of 1(dollar/unit). Thus we estimate its total cost for 2 units to be 4 dollars due to the estimated cost of 1 dollar for the additional unit produced. Now we find that at 2 units of production the marginal cost is 4 (dollars/unit), so we estimate the total cost for 3 units of product to be 4 dollars for the next unit produced, giving an estimate for the total cost of producing 3 units as 8 dollars.
Comparison with The Exact Solution: Notice that in using this method the errors of approximation are compounded as we make approximations based on approximations. The exact solution of this differential equation is found easily since `int x^2 dx = 1/3 x^3 + C`.
Because `3 = f (1) = 1/3 (1)^3 + C` we have that `C = 8/3`. Thus `f (2) = 1/3(2^3) + 8/3 = 16/3 ≈ 5.333` and `f (3) = 1/3(3^3) + 8/3 = 35/3 ≈ 11.667` .
In each case we underestimated the precise value. Can you explain why we should have expected an underestimate from the nature of f '(x)? [Hint: What about the concavity of the graphical interpretation of a solution? What can you say about the acceleration in the dynamic interpretation of a solution?]
If we choose a smaller h at each step, the approximation will be better, but then we will need to process more steps to arrive at an estimate for f(2). Here are the details with dx = h = .1 for estimating f (1.5). [From our exact solution, f (1.5)= (1.5)^{ 3} /3 + 8/3 = 91/24 ≈ 3.7916667].

You can complete this example to estimate f (2) and f (3) yourself. Table 1 shows one way to organize the information to estimate f(2).[Here is a spreadsheet you can use for to complete this example.]
Euler's Method: What is the generalization of this method? It is called Euler's method. As it is described below, keep in mind the examples we have just completed.
Step 1. First, suppose that we are given f (x_{ 0}) = y_{ 0} and we want to estimate a solution to the differential equation f '(x) = P(x,y) where P(x,y) indicates a number that depends (possibly) on both the values of x and y. [In other words P(x,y) is a function of the two variables x and y. You might think of x as time and y as position in the dynamic interpretation so that we have a context where the velocity is determined by time and position. In the economic interpretation you might think of x as units of production and as y the total cost of production so that we have the marginal cost determined by level of production and current total cost.]
Step 2. We choose a value for h = dx and then let x_{ 1} = x_{ 0} + h.
Step 3. Now we estimate f (x_{ 1})
by y_{ 1}= y_{ 0} + P(x_{
0} , y_{ 0})^{ . }h.
We continue by letting x_{ 2} = x_{ 1} + h [= x_{ 0} + 2h] and estimate f (x_{ 2}) by y_{2} = y_{1} + P(x_{1},y_{1})^{ . }h.
In general (see Figure IV.E.4) we let
and estimate f (x_{ }_{k+1}) by
If x_{ 0} = a and x_{ n}
= b then we have that f (b) is estimated by y_{
n} found after n applications of this method.
Since `b = x_n= x_0 + n*h = a + n*h`, if we wish to use this
method to estimate f (b) starting from f (a)
in n steps of equal length, then we use `h = (ba)/n`.
Here is another example to help you understand this further.
Example IV.E.2. Suppose f (1) = 5 and dy/dx = y  x. Estimate f (3) in 4 steps using Euler's method.
Solution: Linking this example to the notation of the previous discussion, we have
So we set `dx = h = (b  a)/n = (3  1) /4 = 1/2 `. Table 2 summarizes the computations for y_{ 1} through y_{ 4} which will be the estimate for f(3). Keep in mind that y_{ k+1} = y_{ k} + P(x_{ k},y_{ k})^{ . }h.
Thus we estimate the value of f (3) with y_{ 4} = 19.1875 .


Note: As with our previous estimation based on the
differential, if greater precision is desired in the estimate we
should use a larger number of steps, and accordingly the value of
h would be smaller. With even a relatively small number of steps
(about 15 or 20) the computation required by Euler's method can
become a rather mind dulling experience.
Here is an opportunity for a programmable calculator, computer program or spreadsheet to demonstrate its labor saving power as well as the power of Euler's method. To give a fairly accurate estimate of the value of f (b) you will want to work with a large number of steps.[You can try using a spreadsheet here.] We'll discuss the accuracy of Euler's method for some cases in the next section. Appendix IV.Y has two sample programs written in BASIC that illustrate the simplicity of Euler's method and how easily it can be translated into computer instructions.
For the differential equations in problems 14, assume y(1) = 2.
(a) Estimate the value of y(3) using Euler's method with n = 4.
Solve the differential equation algebraically and graph the
solution.
Check your results with Sage. [Remember that `dx = h = (b 
a)/n`.]
(b) Draw a graph of the lines in the tangent plane used in Euler's estimation. Compare the estimate with the exact solution. Based on the concavity of the graph explain the quality of the error in your estimate.
(c) Draw a mapping figure that illustrates the dynamic or the
economic interpretations of Euler's method for your work.
Discuss the interpretation of y'' and how it affects the quality
of your estimates.
1. y' = .2 x^{ 3}  2. y' = x  1 
3. dy/dx = .5 x  4. dy/dx = .5 x^{ 2} + 1 
5. y' = 1/x `x != 0`  6. y' = 1/(1+x) `x!=1` 
7. y' = 1/(x^{ 2}+1)  8. y' = x/(x^{ 2} + 1) 
9. dy/dt = 2y  10. dy/dt = y 
11. y' = 2y + x  12. y' = 2x + y 
13. y' = 1/y `y != 0`  14. y' = 2/(1+y) `y!=1` 
15. dy/dx = x^{ 2} + y^{ 2}  16. dy/dt = 1 /(t^{ 2} + y^{ 2}) 
17. y' = x/y `y != 0`  18. y' = y/x `x != 0` 
Euler's method can be extended and generalized in many ways. In problems 2228 we explore extending it to estimate solutions to second order differential equations.
2628. Problems 2325 used 4 steps to estimate y(4). Use the same ideas to estimate the solution to those problems with 8 steps.