General solution vs particular solution

[JamalGross]

I think that particular solutions may come in hand when you already know the basics and just shorten the solution, they are more of shortcuts and I am also using these from time to time. I didn't want to continue this topic I just wanted to express my idea, sorry if it got you angry, also nice to meet you all.

 

[Orodruin]

A point I have repeatedly tried to make.

Well, your starting point in this thread was that there was one and only one particular solution and no other solution could possibly be called a particular solution. This is what we are claiming to be false and arguing against. This

The particular solution is what you (@sdfsfasdfasf) showed, namely, yp(x)=112sin⁡(3x). No other function will satisfy this ODE.

is false and is what started this whole mess. (My emphasis)

Our only disagreement here seems to be that my personal definition of the term "particular solution" is more restrictive than yours, making it at odds with the prevalent definition.

(My emphasis)

If you use your personal definition in teaching, your students will inherit the same disconnection from standard terminology. I believe that this is the issue here.

I look for the basis functions for the homogeneous problem, and then look for a function that solves the nonhomogeneous problem, one that does not contain terms that are any linear combination of the basis functions for the solution of the homogeneous problem.

That's fine of course, but the point is that the method of using a particular solution (any particular solution!) to homogenise the differential equation is not restricted to this way of doing things. You are free to choose any particular solution you can get your hands on, but you have no basis in claiming that it is a preferred particular solution over any other.

These hyperbolic trig functions are also more complicated than what is absolutely necessary.

I could argue that they are about half as complicated as the exponential function. Their definitions in terms of series expansions contain only odd/even terms!

This can be decomposed into cos⁡(2x)+1, which contains a term from the homogeneous problem basis set.

I do not agree with this argument as it can just as well be applied to the function 1 as well. You can rewrite 1 as a sum which contains a cos(2x) apart from the squared cosine. I am not saying it looks more beautiful, but which function is the other plus a homogeneous part is reflexive.

but not ones that I would choose over ex as they would include a term from the homogeneous problem basis set.

But you can just as well write that $e^{x}/2 = \cosh(x) - e^{-x}/2$ so $e^x$ includes a term from the homogeneous problem basis set! This simply is not an argument for one over the other. If you have $f_1(x)$ and $f_2(x)$, which are particular solutions such that $f_2(x) = f_1(x) + h(x)$, where $h(x)$ is a homogeneous solution, then $f_1(x) = f_2(x) - h(x)$, where $-h(x)$ is a homogeneous solution. Do not be fooled by the fact that you may be more familiar with $f_1(x)$ than you are with $f_2(x)$!

 

[sdfsfasdfasf]

This thread did well, I am happy.