Lecture Notes in Differential Equations - Bruce E. Shapiro

36 LESSON 4. LINEAR EQUATIONS
Figure 4.2: The error function, given by equation (4.91).
1
1
Note that because of the way we did the integral we have already taken the
initial condition into account and hence there is no constant C in the result
of our integration.
Now we still do not have a closed formula for the integral on the right but it
is a well-defined function, which is called the error function and written
as
erf(t) = √ 2 ∫ t
e −x2 dx (4.91)
π
The error function is a monotonically increasing S-shaped function that
passes through the origin and approaches the lines y = ±1 as t → ±∞, as
illustrated in figure 4.2. Using equation 4.91 in equation 4.90 gives
and solving for y,
0
ye −t2 = erf(t) (4.92)
y = e t2 erf(t) (4.93)
The student should not be troubled by the presence of the error function
in the solution since it is a well-known, well-behaved function, just like an
exponential or trigonometric function; the only difference is that erf(x) does
not (usually) have a button on your calculator. Most numerical software
packages have it built in - for example, in Mathematica it is represented by
the function Erf[x].
This last example gives us another algorithm for solving a linear equation:
substitute the initial conditions into the integral in step (4) of the algorithm
given on page 28. To see the consequence of this, we return to equation
4.16, but instead take the definite integrals. To have the integration make
sense we need to first change the variable of integration to something other
than t: