In general, finding solutions to these kinds of differential equations can be much more difficult than finding solutions to constant coefficient differential equations. However, if we already know one solution to the differential equation, we can use it to find the second solution. This method is called reduction of order.
Reduction of order requires that a solution already be known.
In order to find a solution to a second order non-constant coefficient differential equation we need to solve a different second order non-constant coefficient differential equation.
However, this isn’t the problem that it appears to be. Because the term involving the v drops out we can actually solve the equation and we can do it with the knowledge that we already have at this point. We will solve this by making a change of variable.
With this change of variable, the equation becomes a linear, first order differential equation that we can solve. This also explains the name of this method. We’ve managed to reduce a second order differential equation down to a first order differential equation.
This method can be used to find second solutions to differential equations. However, this does require that we already have a solution and often finding that first solution is a very difficult task… – Paul Dawkins
The ability to rearrange formulas or rewrite them in different ways is an important skill.
Woke up around midnight last night, with lots of thoughts (the racing thoughts). I called one of my night-owl friends on the phone. My roommate was sleeping so I pulled a kitchen chair out to the hallway and sat there for the entire conversation. We (mostly I) talked until dawn, which is the usual thing that happens whenever I manage to trick somebody into having a phone call or video call with me. (This is probably one of the reasons why nobody wants to talk to me.)
I talked mostly of this analogy I’ve had brewing in my mind for quite some time now, but never got around to putting down on paper (or word processor). It’s partly because I like it so much that I want to deliver it perfectly, and partly because I think it may be somewhat offensive and / or “triggering”, which apparently happens to some people these days a lot. Anyway, I talked about it last night and my friend thought it was great, so I decided I’d at least try to put some of the thoughts down now and keep at it til it sounds the way I want it to.
Nov. 24, 2015 (midnight)
M: I have this analogy about exercise, where I compare it to… well, I compare it to sex.
M: Yeah. Because it’s something that you should be doing because you want to. Because it feels good.
M: It’s supposed to be this natural, pleasurable experience. A healthy body wants to move. We’re not made to be stuck sitting in chairs all day. Children, before they’ve become resigned to being forced to sit still all the time, they love running around, nobody makes them do it. They do it because they want to and it makes them so happy.
F: Yeah, that makes sense.
Me: The thing is, though, after the modern world’s gotten us used to being stationary, it becomes harder and harder to move. Like something that used to be spontaneous and joyful has to be forced.
Me: If sex is a pleasurable activity that requires partners to be in cooperation, healthy movement is a pleasurable activity that requires the mind and body to be in cooperation.
F: *stifles yawn*
Me: Are you sleepy? I’m sorry.
F: No I’m not, go on.
Me: Also, like sex, there’s lots of different ways to do it. For example, I hate running, it’s like torture for me. But I’m okay with biking, and I love the water, so I do those things and it’s good. People have this idea of exercise like it’s just a task and like it’s supposed to be difficult and painful when it’s supposed to be something you look forward to. Not the results, but the movement itself. Just the joy of being present in your body.
F: uh huh.
Me: Forcing yourself to exercise is like forcing your partner to have sex when they don’t want to.
F: Oh my god
Me: Yeah… I’m sorry.
F: No, it’s okay, it makes sense.
Me: Well okay. The thing is, if you want to make things work with somebody and be in a healthy relationship, it wouldn’t be the right thing to just completely stop having sex when you still need it. You have to work through it, experiment, be patient about finding things that are okay with your partner. But that’s what people do when, for example, they try jogging and it hurts and they hate it and they just go back to their TVs and chips. They’re disengaging from the body, essentially they’re giving up on living within their bodies and they just wanna exist within whatever show they’re watching or in virtual chatrooms or wherever they’re putting their minds. While their body just sits or lays there kinda abandoned. It’s crazy because your relationship with your body is the one relationship you’ll never be able to leave as long as you’re alive. Literally.
Are you asleep?
Me: In this analogy, forcing yourself to go to the gym when you really don’t want to would be kind of like building a sex dungeon and just dragging your unwilling partner there for sex.
F: OH MY GOD!
Me: Yeah. I’m sorry.
(There was more but this is what I’ll start with for now)