In this video I go over an example on differential equations and show that the family of functions y = (1 + cet) / (1 - cet), where c is a constant, is a solution to the differential equation y' = 1/2(y2 - 1). The process of proving that it is indeed a solution is to simply take the derivative of the family of functions and ensure that it satisfies the differential equation, which I show that it clearly does.
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Modeling with Differential Equations: Example 1
Show that every member of the family of functions:
is a solution of the differential equation:
Solution
Thus for every value of c, the given function is a solution of the differential equation.
The figure below shows graphs of seven members of the family of solutions:
The differential equation shows that if y ≈ +/- 1 , then y' ≈ 0.
This is shown by the flatness of the graphs near y = 1 and y = -1.