General solution vs particular solution

[sdfsfasdfasf]

Well this post is doing well!
I need some further clarification. For example, in my textbook I am given a 2nd order inhomogeneous differential equation, from which I am able to solve the auxiliary equation, and find a, not the particular solution, and clearly the particular solution I choose cannot have a part of the inhomogeneous solution in it (could someone explain why please!). Now, I am also given some initial conditions, my question to all of you is, what do I call the final answer? It contains no undetermined constants, clearly it solves the differential equation, and it contains a part of the homogeneous solution. Is particular solution a good name for it? I struggle to justify that name. Also, is there only one function f which solves the differential equation and has the required initial condititions? All throughout my textbook it states things like given these (conditions) find the particular solution, that seems to go against everything I have read here.

 

[PeroK]

Yes. However, by "particular solution," I'm considering only functions that aren't also part of the solution to the homogeneous problem.

That has no meaning in general. In this case, I just explicitly added a term in $\sin 3x$, but for a different ODE, it would less obvious where and when you can subtract a solution to the homogeneous equation. In any case, I claim that the particular solution is:

Any function which is a solution to the inhomogeneous ODE.

Whereas, it's not clear what your mathematical definition is. How do you intend to uniquely identify this function from its mathematical properties? You need to specify what is unique about it.

The particular solution to a inhomogeneous ODE is the unique solution, which ...?

Also, the choice of $\cos(3x)$ and $\sin(3x)$ as the basis for your solution space is arbitrary. That basis emerges from a standard technique, but it doesn't make those two functions the only functions with that property. In fact, if we extend things to allow complex solutions, then we have a common alternative in terms of complex exponentials: $e^{3ix}$ and $e^{-3ix}$. Even without complex numbers, there are infinitely many pairs of functions that span the null space.

This is an important point for the the student, as there are many cases in theory and application where the arbitrariness of the functions you choose to represent the null space and the particular solution is an important concept.

Also, just to repeat the quotation from Paul's online notes:

The particular solution is "any solution we can get our hands on". That is very different from your belief that there is a unique choice of particular solution.

 

[PeroK]

Well this post is doing well!

Sorry for the unseemly squabbling.

I need some further clarification. For example, in my textbook I am given a 2nd order inhomogeneous differential equation, from which I am able to solve the auxiliary equation, and find a, not the particular solution, and clearly the particular solution I choose cannot have a part of the inhomogeneous solution in it (could someone explain why please!).

I don't understand what you mean here.

Now, I am also given some initial conditions, my question to all of you is, what do I call the final answer? It contains no undetermined constants, clearly it solves the differential equation, and it contains a part of the homogeneous solution. Is particular solution a good name for it?

As I said above, in Paul's online notes he calls this the "actual" solution. I'm not sure whether looking for standard terminology at this level is a good idea. Authors are going to use different terminology and you are going to have to deal with it.

I might say it is the solution to the IVP (initial value problem).

I struggle to justify that name. Also, is there only one function f which solves the differential equation and has the required initial condititions? All throughout my textbook it states things like given these (conditions) find the particular solution, that seems to go against everything I have read here.

I think that is using "particular" in a different semantic context and should be avoided. As it's already standard terminology for a chosen solution to the general inhomogeneous equation.

Textbook authors are only human and you will find that sort of minor inconsistency here and there.

 

[Orodruin]

Also, the choice of $\cos(3x)$ and $\sin(3x)$ as the basis for your solution space is arbitrary. That basis emerges from a standard technique, but it doesn't make those two functions the only functions with that property. In fact, if we extend things to allow complex solutions, then we have a common alternative in terms of complex exponentials: $e^{3ix}$ and $e^{-3ix}$. Even without complex numbers, there are infinitely many pairs of functions that span the null space.

Here is another example, where the choice of particular solution does not have a clear ”simplest” form:

Consider $y’ + y = e^x$. Since $de^x/dx = e^x$, this has a particular solution $y_p = e^x/2$. However, other particular solutions include $y_p = \sinh(x)$ and $y_p = \cosh(x)$, which are just as viable.

 

[Mark44]

With regard to $y'' + 9y = \frac 1 2 \cos(3x)$:

I claim that the function $$y_p(x) = \frac{(x + 1)}{12}\sin(3x)$$is also a particular solution of the ODE. Can you explain why this is not a particular solution of the ODE?

Sure, this is a particular solution, one that can be decomposed into $\frac x {12}\sin(3x)$ and $\frac 1{12}\sin(3x)$. The former is a solution to the nonhomogeneous DE above, and the latter is a solution of the homogeneous equation.
My whole argument here is why go out of one's way to devise a particular solution that isn't disjoint from the homogeneous problem's solution set?

Also, the choice of $\cos(3x)$ and $\sin(3x)$ as the basis for your solution space is arbitrary. That basis emerges from a standard technique, but it doesn't make those two functions the only functions with that property. In fact, if we extend things to allow complex solutions, then we have a common alternative in terms of complex exponentials: $e^{3ix}$ and $e^{-3ix}$. Even without complex numbers, there are infinitely many pairs of functions that span the null space.

Of course, a basis for the solution space isn't unique, and I don't think I ever said anything to the contrary.

Right, I can't distinguish $x_p$ from $x_p + x_h$

But this is what you are trying to do by claiming that there only exists one particular solution!

So far, the examples of particular solutions that violate my claim appear to be 1) those that are a sum of a viable solution to the nonhomogeneous problem plus a function that is some linear combination of basis functions that are solutions to the homogeneous problem. E.g. $y_p = \frac x {12}\sin(3x) + \frac 1 {12}\sin(3x)$, or 2) functions that are identically equal to some solution of the nonhomogeneous problem. E.g. $y_p = 1$ vs. $y_p = 2\cos^2(x) - \cos(2x)$ (relative to y'' + 4y = 4).

but I can distinguish $x_p$ from $x_h$ by virtue of the fact that $\mathcal L(x_h) = 0$ and $\mathcal L(x_p) = y$. $x_h \in {\rm ker}\mathcal L$ while $x_p \notin {\rm ker}\mathcal L$.

Which is irrelevant here as it is not what is being discussed.

It's relevant to me in the way I approach these kinds of problems; namely, I find a basis of functions that solve the homogeneous problem, and then find a particular solution that is disjoint from that basic set.

Here is another example, where the choice of particular solution does not have a clear ”simplest” form:
Consider $y’ + y = e^x$. Since $de^x/dx = e^x$, this has a particular solution $y_p = e^x/2$. However, other particular solutions include $y_p = \sinh(x)$ and $y_p = \cosh(x)$, which are just as viable.

With trig or hyperbolic trig functions, one can find other functions that are identically equal. The example you gave of $1 = 2\cos^2(x) - \cos(2x)$ falls into this category. Can you argue that the two sides of this equation are actually different functions?

 

[PeroK]

With regard to $y'' + 9y = \frac 1 2 \cos(3x)$:Sure, this is a particular solution, one that can be decomposed into $\frac x {12}\sin(3x)$ and $\frac 1{12}\sin(3x)$. The former is a solution to the nonhomogeneous DE above, and the latter is a solution of the homogeneous equation.
My whole argument here is why go out of one's way to devise a particular solution that isn't disjoint from the homogeneous problem's solution set?

Define "disjoint" in this context.

 

[sdfsfasdfasf]

I have read all of your comments.
I will post an image from my textbook, and would like all of your opinion on it.

Part a) is fine, just differentiate the top expression with respect to t and make the appropriate substitutions. Part b) asks you to find a general solution, so we find a particular solution (namely $x(t)=10$) and add that onto the general solution of the inhomogeneous equation. Part c) is also straightforward (use the top equation to find $y(t)$ by differentiating $x(t)$ and making the appropriate substitution). However in part d) we are given some initial conditions and then asked to find the particular solution. Technically speaking, are the functions we found (such as $x(t) = 10$) not also particular solutions? Clearly what makes the particular solution in part d) different to $x(t) = 10$ is that it matches the required initial conditions.
Thoughts?
My textbook calls the function $x(t) = 10$ the particular integral, and reserves the phrase particular solution for the final answer with no undetermined constants. Found a similar post, will link below. https://www.physicsforums.com/threa...ular-integral-and-particular-solution.667568/

 

[PeroK]

I have read all of your comments.
I will post an image from my textbook, and would like all of your opinion on it.
View attachment 341670
Part a) is fine, just differentiate the top expression with respect to t and make the appropriate substitutions. Part b) asks you to find a general solution, so we find a particular solution (namely $x(t)=10$) and add that onto the general solution of the inhomogeneous equation. Part c) is also straightforward (use the top equation to find $y(t)$ by differentiating $x(t)$ and making the appropriate substitution). However in part d) we are given some initial conditions and then asked to find the particular solution. Technically speaking, are the functions we found (such as $x(t) = 10$) not also particular solutions? Clearly what makes the particular solution in part d) different to $x(t) = 10$ is that it matches the required initial conditions.
Thoughts?

This needs to be in a new thread, IMO.

 

[sdfsfasdfasf]

This needs to be in a new thread, IMO.

It is still under the main topic question "What actually is a particular solution?".

 

[Orodruin]

It's relevant to me in the way I approach these kinds of problems; namely, I find a basis of functions that solve the homogeneous problem, and then find a particular solution that is disjoint from that basic set.

Any function that solves the non-homogeneous equation is a particular solution and therefore is explicitly disjoint from the homogeneous solutions. As we have discussed and shown, there is no uniqueness here. Any particular solution will do.

This is very standard and a basic fact about linear differential equations. You simply cannot make the distinction you are attempting to do in singling out a specific particular solution. As @PeroK said, you would have to mathematically be able to define the properties of such a unique function. ”Does not contain a part that is on the homogeneous form” will not do as previously shown as functions may be rewritten on different forms where a homogeneous solution term pops out. So I ask you again to provide your actual mathematical definition.

For the solution process, it does not matter which particular solution you choose, but teaching that there is only one unique particular solution borders on spreading misinformation in my book.

that is disjoint from that basic set.

Again, this is not well defined. I know what you are trying to say, but things simply do not work like that and unless you specify the mathematical properties of the function you want to choose, it will remain ambiguous. There is also nothing stopping you from adding an arbitrary fixed homogeneous solution to a particular solution and get a new particular solution that you can work with just as well. In fact, the preferable way to solve these problems should be to first find the particular solution, which reduces the remaining problem to solving the homogeneous part, which is more transparent in my opinion.

With trig or hyperbolic trig functions, one can find other functions that are identically equal. The example you gave of 1=2cos2⁡(x)−cos⁡(2x) falls into this category. Can you argue that the two sides of this equation are actually different functions?

Obviously they are the same function, but you are missing the point. The point is that 1 can be rewritten in a manner such that it is not disjoint from the homogeneous (your words).

 

[Orodruin]

However in part d) we are given some initial conditions and then asked to find the particular solution.

No. You are asked to find the particular solution that satisfies <<conditions>>. Here the <<conditions>> (typically boundary/initial conditions) single out one of the many particular solutions.

 

[Mark44]

Define "disjoint" in this context.

-- is not a sum that contains at least one function that is a linear combination of the homogeneous equation's basis set.

 

[Mark44]

In fact, the preferable way to solve these problems should be to first find the particular solution, which reduces the remaining problem to solving the homogeneous part, which is more transparent in my opinion.

I'm glad you added the "in my opinion part." The way I was taught and later taught this subject, was to find the solution to the homogeneous problem, and then find a particular solution.
One technique that I found to be fruitful was the so-called method of annihilators.
Using your example again, y'' + 4y = 4, this can be written as $(D^2 + 4)y = 4$.
Since the D (or differentiation) operator annihilates any constant the nonhomogeneous problem above can be rewritten as a higher order homogeneous equation as $(D^3 + 4D)y = D(4) = 0$.
Factoring the left side gives $D(D + 2i)(D - 2i)y = 0$, so a basic set for this new equation is $\{1, e^{2ix}, e^{-2ix}\}$. De Moivre's formula allows us to change this set to $\{1, \cos(2x), \sin(2x)\}$. Of course the latter two functions are in the kernel of the homogeneous equation, so I see little point in including them as part of a particular solution.
A particular solution can be found by determining which multiple of the basis function 1 satisfies the nonhomogeneous equation. Letting $y_p = A \cdot 1$, we have $y_p'' + 4 y_p = 4$. Since $y_p'' = 0$, we have $4y_p = 4 \Rightarrow y_p = 1$.
This leads us to a general solution of the original nonhomogeneous ODE being $y_g = c_1\cos(2x) + c_2\sin(2x) + 1$, where the two constants can be determined if initial conditions are included.

 

[Orodruin]

-- is not a sum that contains at least one function that is a linear combination of the homogeneous equation's basis set.

But, as has been shown with several examples, this does not uniquely identify one of the particular solutions. It is therefore not a mathematically rigorous way of singling out ome of the many particular solutions.

More to the point: There is absolutely nothing wrong with using a particular solution that has a term that is a homogeneous solution. You will get different constants when you adapt the general solution to whatever initial/boundary conditions you may have, but the final solution will be the same.

This is where teaching that there is only one permissible particular solution goes wrong. What is worse, you might mark down someone who used $\sinh(x)$ instead of $\cosh(x)$ or $e^x/2$ as their particular solution when in reality they did something completely fine.

One should also note that a particular solution that also satisfies the initial conditions is what is required to solve a diffential equation. It is explicitly what the OP’s problem asks for.

 

[Mark44]

More to the point: There is absolutely nothing wrong with using a particular solution that has a term that is a homogeneous solution.

Other than it is a waste of effort? The term that is a homogeneous solution vanishes when it is substituted in the DE. That's really my main point.

You will get different constants when you adapt the general solution to whatever initial/boundary conditions you may have, but the final solution will be the same.

If we both arrive at the same solution, then what exactly is the problem?

This is where teaching that there is only one permissible particular solution goes wrong. What is worse, you might mark down someone who used $\sinh(x)$ instead of $\cosh(x)$ or $e^x/2$ as their particular solution when in reality they did something completely fine.

This is something I strove hard never to do. When I was marking homework or exams, and someone had an answer that looked different from the answer sheet I had prepared, but was equivalent to mine, I never marked them down. Never.

 

[Orodruin]

Other than it is a waste of effort? The term that is a homogeneous solution vanishes when it is substituted in the DE. That's really my main point.

There really is no point here. As has been shown repeatedly and with different examples, it is not always directly obvious that two different homogeneous solutions have terms from the homogeneous solution.

They must of course differ by a homogeneous solution, but they are always equally permissible and teaching that there is only a single particular solution is simply wrong. All solutions where you fix the arbitrary constants are particular solutions by definition. That you choose to use a specific one out of those is a matter of your preferred way to solve a problem, not one of making a particular solution unique

This is something I strove hard never to do. When I was marking homework or exams, and someone had an answer that looked different from the answer sheet I had prepared, but was equivalent to mine, I never marked them down. Never.

What if they only solved the problem partially and used a different particular solution? The final answer is not everything.

If we both arrive at the same solution, then what exactly is the problem?

Nothing. It is you who seems to have a problem with there being more than one particular solution.

 

[PeroK]

-- is not a sum that contains at least one function that is a linear combination of the homogeneous equation's basis set.

Okay, so if we have explicit functions, you can look at the function and check that it meets this criteria. But, what if we have a particular solution in theory: $y_p(x)$. How do we confirm that we have the unique particular solution? Suppose you say you have found the unique particular solution. How can you prove it?

I would like to see your proof that $y_p(x)$ is not a sum that contains at least one function that is a linear combination of the homogeneous equation's basis set. I.e. a proof that $y_p(x)$ can be determined uniquely?

 

[Orodruin]

I would like to see your proof that $y_p(x)$ is not a sum that contains at least one function that is a linear combination of the homogeneous equation's basis set. I.e. a proof that $y_p(x)$ can be determined uniquely?

And I would like to see a valid method for squaring the circle while we are at it ...

 

[Mark44]

As has been shown repeatedly and with different examples, it is not always directly obvious that two different homogeneous solutions have terms from the homogeneous solution.

I think you got in a hurry here and wrote something different from what you meant.

They must of course differ by a homogeneous solution, but they are always equally permissible...

A point I have repeatedly tried to make.

... and teaching that there is only a single particular solution is simply wrong.

Okay, okay, please stop beating this dead horse! 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. So mea culpa. In my usual approach to these kinds of problems, 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.

Regarding the examples given so far:
$y'' + 9y = \frac 1 2 \cos(2x)$
Homogeneous problem basis set: $\{\sin(3x), \cos(3x) \}$
For the nonhomogeneous problem I would consider multiples of $\{x\sin(3x), x\cos(3x) \}$
I will admit that $\frac{x + 1}{12}\sin(3x)$ is a particular solution, but not one that I would choose, as it can be decomposed into a sum with a term from the basis set
($\frac 1{12}\sin(3x)$) for the homogeneous problem. It's also more complicated than what is absolutely necessary.

$y'' + 4y = 4$
Homogeneous problem basis set: $\{\sin(2x), \cos(2x) \}$
For the nonhomogeneous problem I would consider a constant function; i.e., a multiple of 1.
I will admit that $2\cos^2(x)$ is a particular solution, but not one I would choose. This can be decomposed into $\cos(2x) + 1$, which contains a term from the homogeneous problem basis set. It too is also more complicated than what is absolutely necessary.
I also would not consider any function that is identically equal to some constant, such as $\sin^2(kx) + \cos^2(kx)$, $\sec^2(kx) - \tan^2(kx)$, etc.

$y' + y = e^x$
Homogeneous problem basis set: $\{e^{-x}\}$
I will admit that $\sinh(x)$ and $\cosh(x)$ are potentially particular solutions, but not ones that I would choose over $e^x$ as they would include a term from the homogeneous problem basis set. These hyperbolic trig functions are also more complicated than what is absolutely necessary.

 

[Mark44]

Can we be done now?

 

[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.