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: https://courses.physicswithelliot.com/notes-sign-up Sign up for my newsletter for additional physics lessons: https://www.physicswithelliot.com/sign-up 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! Related videos: All about the simple harmonic oscillator, and why it's so important: https://www.youtube.com/watch?v=bmGqhM-tUk4 The Fourier transform, with applications to quantum mechanics: https://www.youtube.com/watch?v=W8QZ-yxebFA The math and physics of Taylor series: https://www.youtube.com/watch?v=HQsZG8Yxb7w 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 If you find the content I’m creating valuable and would like to help make it possible for me to continue sharing more, please consider supporting me! You can make a recurring contribution at https://www.Patreon.com/PhysicsWithElliot, or make a one time contribution at https://www.physicswithelliot.com/support-me. Thank you so much! 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 https://www.physicswithelliot.com/.