i dont get why we're allowed to assume a solution is of the form u(t)*y1(t).... how do we know its a function multiple of a homogeneous solution? what if it cant be written that way?
For any particular solution y_p, we could always write it as (y_p/y_1)*y_1, and there's our function multiple on the left. For constant coefficients, the complementary solutions are always exponentials and therefore nonzero. The only possible exceptions would be weird nonconstant coefficient cases where both complementary solutions are zero at some point, but the particular solution is nonzero. I can't think of an example of that off the top of my head
I'm finished with this series for now, but I've also done some other differential equations videos! Here are two playlists to look at: Differential equation battles Differential Equation Battles: th-cam.com/play/PLug5ZIRrShJFPgeyYpfKf2lazeZ3axKqX.html Laplace transforms Laplace Transforms: th-cam.com/play/PLug5ZIRrShJER_zQ-IVVefmsh9vZHwGnv.html
Brilliant , thanks for clarification of theory behind this method to solve second order equation
Didn't expect to get it first try. Was surprised. You're a pro!
i dont get why we're allowed to assume a solution is of the form u(t)*y1(t).... how do we know its a function multiple of a homogeneous solution? what if it cant be written that way?
For any particular solution y_p, we could always write it as (y_p/y_1)*y_1, and there's our function multiple on the left.
For constant coefficients, the complementary solutions are always exponentials and therefore nonzero. The only possible exceptions would be weird nonconstant coefficient cases where both complementary solutions are zero at some point, but the particular solution is nonzero. I can't think of an example of that off the top of my head
@@MuPrimeMath oh hmm, yeah i guess. cool
Wow, you're amazing! I didn't understand why we do all of this, but now I finally get it.
I love your explanation
amazing work
Wow. This is amazing. Also, may I ask you to explain this method but with higher order ? And btw, thank you so much for making this video!
When is part 10 coming?
I'm finished with this series for now, but I've also done some other differential equations videos! Here are two playlists to look at:
Differential equation battles Differential Equation Battles: th-cam.com/play/PLug5ZIRrShJFPgeyYpfKf2lazeZ3axKqX.html
Laplace transforms Laplace Transforms: th-cam.com/play/PLug5ZIRrShJER_zQ-IVVefmsh9vZHwGnv.html
Cool