Differential equations are hard! But these 5 methods will enable you to solve all kinds of equations that you'll encounter throughout your physics studies. Get the notes for free here: courses.physicswithelliot.com...
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Almost every physics problem eventually comes down to solving a differential equation. But differential equations are really hard! Fortunately, there are powerful tools for tackling them, and in this video I'll introduce you to five of them: substituting an ansatz, using energy conservation, making a series expansion, using the Laplace transform, and finally using Hamilton's equations, which give a new way to visualize the solution as what's called a flow on phase space, as well as a way to solve an equation with a matrix exponential.
We'll see how they all work using one of the most important differential equations in physics: the F=ma equation for a simple harmonic oscillator, or in other words a block attached to a spring. You certainly don't need crazy powerful tools to solve such a simple equation, but seeing how they work in a simple problem will help prepare you for the harder problems you'll inevitably meet later on in physics!
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0:00 Introduction
2:20 The equation
4:01 1: Ansatz
9:10 2: Energy conservation
14:17 3: Series expansion
18:23 4: Laplace transform
22:41 5: Hamiltonian Flow
26:48 Matrix Exponential
29:31 Wrap Up
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About me:
I’m Dr. Elliot Schneider. I love physics, and I want to help others learn (and learn to love) physics, too. Whether you’re a beginner just starting out with your physics studies, a more advanced student, or a lifelong learner, I hope you’ll find resources here that enable you to deepen your understanding of the laws of nature. For more cool physics stuff, visit me at www.physicswithelliot.com.