Implicit Solution To Differential Equation
2.2. Finding Implicit Solutions
Separating differential equations into x and y parts is fine; it can besides be quite helpful. Yet sometimes you merely can't come upward with a groovy y = f(x) solution, no matter how hard y'all try. For case, what if y'all encounter a differential equation like this ane?
You lot can multiply both sides by dx to get
(1 − y 2) dy = x 2 dx
As y'all can see, this is a separable differential equation. Integrating both sides gives you
Doesn't exactly look easy to write in the grade y = f(x), does information technology? That's because it'south an implicit solution, also known as whatever solution you can't write like y = f(x). (Solutions that can exist written the piece of cake way are considered explicit solutions.)
Although finding implicit solutions can be useful, sometimes you end up having to resort to numerical methods on a computer to convert them into the standard y = f(x) form. On the other mitt, finding an implicit solution is occasionally the very best you can do.
Check out how to solve the post-obit implicit solution problem then try your manus at a few that are just like it.
NOTE
EXAMPLE
Q. Find an implict solution to this differential equation:
A.
-
Multiply both sides by dx:
(y − y 2) dy −
Implicit Solution To Differential Equation,
Source: https://www.oreilly.com/library/view/differential-equations-workbook/9780470472019/9780470472019_finding_implicit_solutions.html
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