1.8: Substitution
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- Apply substitution to simplify and solve non-separable differential equations.
- Solve Bernoulli equations using the substitution
Just as when solving integrals, one method to try is to change variables to end up with a simpler equation to solve.
Substitution
The equation
is neither separable nor linear. What can we do? How about trying to change variables, so that in the new variables the equation is simpler. We use another variable
We need to figure out
In other words,
So
for some constant
Now we need to “unsubstitute” to obtain
and also the two solutions
Note that
Substitution in differential equations is applied in much the same way that it is applied in calculus. You guess. Several different substitutions might work. There are some general things to look for. We summarize a few of these in Table 1.8.1.
When you see | Try substituting |
---|---|
Usually you try to substitute in the “most complicated” part of the equation with the hopes of simplifying it. The above table is just a rule of thumb. You might have to modify your guesses. If a substitution does not work (it does not make the equation any simpler), try a different one.
Bernoulli Equations
There are some forms of equations where there is a general rule for substitution that always works. One such example is the so-called Bernoulli equation.
This equation looks a lot like a linear equation except for the
Solve
Solution
First, the equation is Bernoulli
In other words,
Now the equation is linear. We can use the integrating factor method. In particular, we use Formula (1.6.2). Let us assume that
We now plug in to Formula (1.6.2)
Note that the integral in this expression is not possible to find in closed form. As we said before, it is perfectly fine to have a definite integral in our solution. Now “unsubstitute”
Remember Formula (1.6.2) when we solve
if we let
then
Homogeneous Equations
Another type of equations we can solve by substitution are the so-called homogeneous equations. Suppose that we can write the differential equation as
Here we try the substitutions
We note that the equation is transformed into
Hence an implicit solution is
Solve
Solution
We put the equation into the form
which has a solution
We unsubstitute
We want
Thus
Footnotes
[1] There are several things called Bernoulli equations, this is just one of them. The Bernoullis were a prominent Swiss family of mathematicians. These particular equations are named for Jacob Bernoulli (1654–1705).
Contributors and Attributions
- Jiří Lebl (Oklahoma State University).These pages were supported by NSF grants DMS-0900885 and DMS-1362337.