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300s School/301. ME 2016 - Nonlinear Dynamical Systems 1/2024-09-09 Frameworks and Review.md

@@ -14,3 +14,18 @@ Well, a couple of places...
 2. External fields
 3. Material properties
 So we're stuck with them. But how do we deal with noninearities?
+
+## A nonlinear equation
+$$ \dot{x} = \frac{dx}{dt} = 1-2\cos x$$
+How do you solve this? You can't use Laplace, you can't separate...
+*insert very long expression that Bajaj wrote.*
+Getting an analytical solution can be a PITA to obtain. For this reason:
+**The general case is that nonlinear equations are unsolvable.**
+This doesn't mean we can't learn things. We can describe these systems *qualitatively*.
+
+Really our options come down to:
+- Solve exactly (Not likely to happen)
+- Solve numerically
+- Analyze qualitatively (~geometrically)
+- Solve an approximation to the problem
+We mix and match these approaches.