In this video I go over the concept of separable equations which are a type of differential equations that are possible for us to solve explicitly. A separable equation is a first-order differential equation in which both the independent and dependent variable can be separated in such a way as to write the derivative as being equal to a function of the first variable multiplied by a function of the other variable. This allows the differential equation to be "separated" in such a way that all the variables of one type are on side of the equation and the others on the other side. This gives the opportunity to apply an integral to both sides of the equation and thus serves as implicitly showing the solution of the differential equation. Depending on the complexity of the separable equation, it may be possible to solve for the solution directly.
This method of separable equations was first used by James Bernoulli and later general derived in a paper by his brother John Bernoulli. In fact the Bernoulli family is one of the most famous families in history and have a major role in shaping the foundation of mathematics. Thus, I have also gone over a brief history on the Bernoulli family in this video, so make sure to watch it!
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Differential Equations: Separable Equations
We have looked at first-order differential equations from a geometric point of view (direction fields) and from a numerical point of view (Euler's method).
What about the symbolic point of view?
It would be nice to have an explicit formula for a solution of a differential equation.
Unfortunately, that is not always possible.
But there are certain types of differential equations that can be solved explicitly.
In this video I will take a look at one specific type.
A separable equation is a first-order differential equation in which the expression for dy/dx can be factored as a function of x times a function of y.
In other words, it can be written in the form:
The name separable comes from the fact that the expression on the right side can be separated into a function of x and a function of y.
Equivalently, if f(y) ≠ 0, we could write:
To solve this equation, we first rewrite it in the differential form:
so that all y's are on one side of the equation and all x's are on the other side.
Then we integrate both sides of the equation:
This equation defines y implicitly as a function of x.
In some cases we may be able to solve for y in terms of x.
We can justify this procedure by working backwards by applying the Chain Rule:
Brief (Mainstream) History
This technique for solving separable equations was first used by James Bernoulli (in 1960) in solving a problem about pendulums and by Leibniz (in a letter to Huygens in 1691).
John Bernoulli explained the general method in a paper published in 1694.
James Bernoulli (also known as Jacob or Jacques) lived from 1655 to 1705 and was one of the many prominent mathematicians in the Bernoulli family.
- He was born in Switzerland and was a major contributor to the earlier development in the field of probability.
John Bernoulli (brother of James) lived from 1667 to 1748 and also was a prominent mathematician in the Bernoulli family.
- He was also born in Switzerland and was a major contributor to infinitesimal calculus.
- He also educated the famous and gifted mathematician Leonhard Euler.
The Bernoulli family were famous for having produced many notable artists, scientists, mathematicians, but three in particular (James, John, and Nicolaus) that became cornerstones in the foundation of mathematics.