Ansatz for particular solution - Inhomogenous diff equation

[hari123]

Homework Statement

I am trying to solve 2nd order in-homogeneous equation of y(x) (given below). I was able to get the Homogeneous solution , but i am not able to create the Ansatz for the particular solution.

It would be really helpful if anyone suggests an Ansatz for this equation and also the approach in which he/she got to this Ansatz using the Standard available Ansatz types.

Homework Equations

y'' + (3/b)*y' = 12*(x/b³), where b = x +2

The Attempt at a Solution

I was able to get the Homogeneous solution as,
y_hom = -C1/(2*b2) + C2 (C1,C2 are integration constants). (I have attached the derivation below)

 

[Antarres]

Hello,

You can notice that in order for the left side of the equation to have the same form as the right side, you can choose the 1st derivative to be:

y' = (Ax + B) / (x+2)²

Proceeding to get the second derivative, and then solving for A and B would give you the appropriate particular solution, I think.

Edit: And well, I don't know any standard Ansatz methods, I just guess by intuition. Basically you look at the form of the equation and try to fit one term you don't have with the general form, and then look if other terms will fit, too. So this was actually just my first guess, but I think it would work.

 

[Ray Vickson]

Homework Statement

I am trying to solve 2nd order in-homogeneous equation of y(x) (given below). I was able to get the Homogeneous solution , but i am not able to create the Ansatz for the particular solution.

It would be really helpful if anyone suggests an Ansatz for this equation and also the approach in which he/she got to this Ansatz using the Standard available Ansatz types.

Homework Equations

y'' + (3/b)*y' = 12*(x/b³), where b = x +2

The Attempt at a Solution

I was able to get the Homogeneous solution as,
y_hom = -C1/(2*b2) + C2 (C1,C2 are integration constants). (I have attached the derivation below)

View attachment 223408

Why bother with an "ansatz"? You can just write down and exact solution. Letting ##z = y'## gives you a DE if the form ##z' + p(x) z = q(x)##, which can be solved using the classical technique of an "integrating factor".

 

[hari123]

Hello,

You can notice that in order for the left side of the equation to have the same form as the right side, you can choose the 1st derivative to be:

y' = (Ax + B) / (x+2)²

Proceeding to get the second derivative, and then solving for A and B would give you the appropriate particular solution, I think.

Edit: And well, I don't know any standard Ansatz methods, I just guess by intuition. Basically you look at the form of the equation and try to fit one term you don't have with the general form, and then look if other terms will fit, too. So this was actually just my first guess, but I think it would work.

Thank you for your reply @Antarres .

I have tried using y' = (Ax + B) / (x+2)² and I was able to get the desired particular solution.

But i have a small doubt about this idea, in the line " try to fit one term with the general form" of your reply.
What do you mean with "general form" ? Do you mean the Non-homogeneous term (right hand side term) ?

 

[hari123]

Why bother with an "ansatz"? You can just write down and exact solution. Letting ##z = y'## gives you a DE if the form ##z' + p(x) z = q(x)##, which can be solved using the classical technique of an "integrating factor".

Thank you for your reply @Ray Vickson

I am not quite familiar with the Integrating factor technique. I will go through it and then will try to solve the equation.

 

[Antarres]

Thank you for your reply @Antarres .

I have tried using y' = (Ax + B) / (x+2)² and I was able to get the desired particular solution.

But i have a small doubt about this idea, in the line " try to fit one term with the general form" of your reply.
What do you mean with "general form" ? Do you mean the Non-homogeneous term (right hand side term) ?

General form means like, rational function, trigonometric function, or other function of certain form. For example here, you have x/(x+2)³, so you can guess that your function that would fit in the place of y' would have to have 1/(x+2)², because coefficient next to it has the form of 1/(x+2). And it would also have to have a linear function in the numerator, since that's what you're missing, so that's what was my first guess. Generally, you can develop this type of intuition with a bit of practice in solving ODEs of this type. So a form of the function is like generally 'what it looks like', is it a sine, is it a cosine, a product of elementary functions or an elementary function of some type. You can't really have general algorithm with this way of solving, but it is useful if you manage to develop this intuition, since you might be able to use it in problems which you haven't seen before, so you wouldn't know what to try there, but you can make educated guesses. Hope that helps :)

And yes, I mean nonhomogenous term in the case when you're looking for particular solution. But like just the form of nonhomogenous term can give you some clue about what to try when you're guessing the particular solution, so you cut out all the constants and just focus on what the functional law of that term is.

 

[hari123]

General form means like, rational function, trigonometric function, or other function of certain form. For example here, you have x/(x+2)³, so you can guess that your function that would fit in the place of y' would have to have 1/(x+2)², because coefficient next to it has the form of 1/(x+2). And it would also have to have a linear function in the numerator, since that's what you're missing, so that's what was my first guess. Generally, you can develop this type of intuition with a bit of practice in solving ODEs of this type. So a form of the function is like generally 'what it looks like', is it a sine, is it a cosine, a product of elementary functions or an elementary function of some type. You can't really have general algorithm with this way of solving, but it is useful if you manage to develop this intuition, since you might be able to use it in problems which you haven't seen before, so you wouldn't know what to try there, but you can make educated guesses. Hope that helps :)

And yes, I mean nonhomogenous term in the case when you're looking for particular solution. But like just the form of nonhomogenous term can give you some clue about what to try when you're guessing the particular solution, so you cut out all the constants and just focus on what the functional law of that term is.

Yup @Antarres , This is really Informative . Thank you