# inhomogeneous ODE 2. Order - a big problem

• Jul 17th 2008, 11:02 PM
Marine
inhomogeneous ODE 2. Order - a big problem
Hello to everybody!

I'm trying to solve the following ODE:

y"(t) -5·y'(t) +6·y(t) = 2·tan (2·t)

here's what I have till now:

(D^2-5D+6)y = 0 is the homogeneous part :)

y_h = A*e^(2t) + B*e^(3t)

What can I do for the particular solution?
In general I know 2 methods - the ansatz with the characteristic polynomial when the inhomogeneous part is a function of the form: a*e^(bx) (a, b are complex numbers) and the Laplace-transform.

Unfortunatelly neither helps me finding y_p and I know it does exist, cuz Mathematica has given it ;)

I really hope you could help me :) If the solution is too difficult to be done manually, then I would be glad to see it described with words or some general formulae.

best wishes and 10x
• Jul 17th 2008, 11:52 PM
mr fantastic
Quote:

Originally Posted by Marine
Hello to everybody!

I'm trying to solve the following ODE:

y"(t) -5·y'(t) +6·y(t) = 2·tan (2·t)

here's what I have till now:

(D^2-5D+6)y = 0 is the homogeneous part :)

y_h = A*e^(2t) + B*e^(3t)

What can I do for the particular solution?
In general I know 2 methods - the ansatz with the characteristic polynomial when the inhomogeneous part is a function of the form: a*e^(bx) (a, b are complex numbers) and the Laplace-transform.

Unfortunatelly neither helps me finding y_p and I know it does exist, cuz Mathematica has given it ;)

I really hope you could help me :) If the solution is too difficult to be done manually, then I would be glad to see it described with words or some general formulae.

best wishes and 10x

Try using variation of parameters: Method of variation of parameters
• Jul 18th 2008, 04:04 AM
Marine
thanks, that helped me a lot :)

but there is an integral of the Form:

Int[tanx*e^x]dx

which must be solved in the 4th step and I failed :(

Does anyone have any suggestions for an analytical solution?
• Jul 19th 2008, 08:58 PM
mr fantastic
Quote:

Originally Posted by Marine
thanks, that helped me a lot :)

but there is an integral of the Form:

Int[tanx*e^x]dx

which must be solved in the 4th step and I failed :(

Does anyone have any suggestions for an analytical solution?

This integral has no solution in terms of a finite number of elementary functions. You might need to check your work. Perhaps post what you've done.
• Jul 19th 2008, 09:21 PM
Chris L T521
Quote:

Originally Posted by Marine
Hello to everybody!

I'm trying to solve the following ODE:

y"(t) -5·y'(t) +6·y(t) = 2·tan (2·t)

here's what I have till now:

(D^2-5D+6)y = 0 is the homogeneous part :)

y_h = A*e^(2t) + B*e^(3t)

What can I do for the particular solution?
In general I know 2 methods - the ansatz with the characteristic polynomial when the inhomogeneous part is a function of the form: a*e^(bx) (a, b are complex numbers) and the Laplace-transform.

Unfortunatelly neither helps me finding y_p and I know it does exist, cuz Mathematica has given it ;)

I really hope you could help me :) If the solution is too difficult to be done manually, then I would be glad to see it described with words or some general formulae.

best wishes and 10x

I'm just throwing this idea out there, but do you know how to solve DE's using Power Series?? I think that may help here [someone correct me if I'm wrong]

--Chris
• Jul 19th 2008, 10:26 PM
Mathstud28
Quote:

Originally Posted by Chris L T521
I'm just throwing this idea out there, but do you know how to solve DE's using Power Series?? I think that may help here [someone correct me if I'm wrong]

--Chris

I do not think this would be a good idea. Have you ever seen the Maclaurin series for \$\displaystyle \tan(x)\$?
• Jul 19th 2008, 10:33 PM
Chris L T521
Quote:

Originally Posted by Mathstud28
I do not think this would be a good idea. Have you ever seen the Maclaurin series for \$\displaystyle \tan(x)\$?

Now that you mention that...yes I have...its not that pretty...(Shake)

--Chris
• Jul 19th 2008, 10:42 PM
Marine
'This integral has no solution in terms of a finite number of elementary functions. You might need to check your work. Perhaps post what you've done.':

hmmm, this function I have not seen yet. But I suppose I could just leave it as an integral in that case. Btw, what's its name?

' but do you know how to solve DE's using Power Series?? I think that may help here': - unfortunately I don't (Doh) .

What I know about ODEs I have learnt by myself - from scripts and I-net sites and from you tube movies. (I just decided to take a look at them in preparation of my study in physics I am going to start in october :). To my surprise this matter appeared to be somehow imperfect (from a point of view of a high school student :))
• Jul 20th 2008, 07:51 PM
mr fantastic
Quote:

Originally Posted by Marine
Hello to everybody!

I'm trying to solve the following ODE:

y"(t) -5·y'(t) +6·y(t) = 2·tan (2·t)

[snip]

What can I do for the particular solution?

[snip]

Unfortunatelly neither helps me finding y_p and I know it does exist, cuz Mathematica has given it ;)

[snip]

What particular solution does mathematica give?
• Jul 20th 2008, 09:00 PM
Marine
I meant it gives a solution to the problem in general, which means there must be a particular solution. I checked in the doc. centre but couldn't find a command which gives only a particular solution.
• Jul 20th 2008, 09:45 PM
mr fantastic
Quote:

Originally Posted by Marine
I meant it gives a solution to the problem in general, which means there must be a particular solution. I checked in the doc. centre but couldn't find a command which gives only a particular solution.

What is the solution it gives to the problem in general?
• Jul 20th 2008, 10:02 PM
Chris L T521
Quote:

Originally Posted by mr fantastic
What particular solution does mathematica give?

http://img.photobucket.com/albums/v4.../DiffEqTan.jpg

--Chris
• Jul 21st 2008, 02:23 AM
Marine
seems pretty complex, doesn't it? (Surprised)

nevertheless, it has an answer and the thread above shows the way to solve it :)

but how do you find a general solution to an ODE of the form:

y''[x]+p[x]*y'[x]+q[x]*y[x]==0

???

in the thread by mr fantastic in his 1st reply, this form is considered as the general form of a linear homogeneous 2nd order ODE (which it is), but they don't teach how to find the solution hmmm (Doh)

(they only show the way with the char. polynomial using the disposition p[x] and q[x] are constant)

so, does anyone of you have an idea?
• Jul 21st 2008, 04:53 AM
mr fantastic
Quote:

Originally Posted by Marine
seems pretty complex, doesn't it? (Surprised)

nevertheless, it has an answer and the thread above shows the way to solve it :)

but how do you find a general solution to an ODE of the form:

y''[x]+p[x]*y'[x]+q[x]*y[x]==0

???

in the thread by mr fantastic in his 1st reply, this form is considered as the general form of a linear homogeneous 2nd order ODE (which it is), but they don't teach how to find the solution hmmm (Doh)

(they only show the way with the char. polynomial using the disposition p[x] and q[x] are constant)

so, does anyone of you have an idea?

Your inexperience with differential equations is evident. The method of solving this DE depends on what p(x) and q(x) are. It may be that it's not even possible to get an analytic solution without inventing a brand new function.

By the way, given the output of Mathematica, I don't understand why you were trying to find an analytic form for the particular solution. Clearly there was not one in terms of functions you were familiar with.

For many of the types of differential equations you'll meet in physics (especially quantum mechanics) you'll use power series methods to solve them. And meet new functions as a result (Legendre polynomials, Laguerre polynomials, hermite polynomials etc).

You would be best served going to a traditional textbook and carefully working through it.
• Jul 21st 2008, 05:42 AM
Marine
Quote:

Originally Posted by mr fantastic
Your inexperience with differential equations is evident. The method of solving this DE depends on what p(x) and q(x) are. It may be that it's not even possible to get an analytic solution without inventing a brand new function.

By the way, given the output of Mathematica, I don't understand why you were trying to find an analytic form for the particular solution. Clearly there was not one in terms of functions you were familiar with.

For many of the types of differential equations you'll meet in physics (especially quantum mechanics) you'll use power series methods to solve them. And meet new functions as a result (Legendre polynomials, Laguerre polynomials, hermite polynomials etc).

You would be best served going to a traditional textbook and carefully working through it.

At the moment I'm searching for a book about DEs but it's not that easy finding one here. Such highly sophisticated math books have a very limited circulation, which makes them difficult to find. :(

I received a Russian math book from my father which deals with most of the problems, however the print is dated to 1960 :D

So the way I try to learn solving them is via Internet, some you tube lecture movies and scripts or special sites.

What's left is just wait till the university, but this option does not appeal to me very much :)