Back when I was taking differential equations, some thirty-plus years ago, it wasn't clear to me what the deal with the homogeneous solution was. Since then I have found clarity. Think of a normal antiderivative, and how there's always a "+ C" that you slap on at the end as a matter of habit. That "+ C" is a homogeneous solution: it's a term that you have to include because, if you do the derivative again, the term would go to zero, so it can be part of any solution. And that's how homogeneous solutions to these differential equations work, they're like the "+ C" but a lot more complicated. I should demonstrate how "+ C" is the homogeneous solution. Let's consider the equation dy/dx = cosx, and of course we all know that the antiderivative is "sinx + C". Another way to approach it is that it's a differential equation, with y' = cosx. The homogeneous solution is "C", it's what we get when we try to solve "y' = 0". It's in solving for the particular solution that we get the "sinx" part of the solution.
Not sure if anyone told you but thank you for making these videos!!!
Back when I was taking differential equations, some thirty-plus years ago, it wasn't clear to me what the deal with the homogeneous solution was. Since then I have found clarity. Think of a normal antiderivative, and how there's always a "+ C" that you slap on at the end as a matter of habit. That "+ C" is a homogeneous solution: it's a term that you have to include because, if you do the derivative again, the term would go to zero, so it can be part of any solution. And that's how homogeneous solutions to these differential equations work, they're like the "+ C" but a lot more complicated.
I should demonstrate how "+ C" is the homogeneous solution. Let's consider the equation dy/dx = cosx, and of course we all know that the antiderivative is "sinx + C". Another way to approach it is that it's a differential equation, with y' = cosx. The homogeneous solution is "C", it's what we get when we try to solve "y' = 0". It's in solving for the particular solution that we get the "sinx" part of the solution.
I Respect the fact that you add information which you forgot to mention.
Yeah, You Are So Lazy
Thank you for the kind words, Tom