Expert in ODE.... please help me how to get this answer??

Question

the question is:-
y''=3y'+2y+sinh(x)

The answer given is:-
y=C2exp[1/2(3+surd(17)x] + C1exp[-1/2(-3+surd(17)x] + 1/272(-10surd(17)-34)exp[-1/2(-3+surd(17))x]cosh[1/2(-5+surd(17)x] - 1/272{-17exp(xsurd(17)sinh[1/2(1+surd(17))x]+10sinh[1/2(-5+surd(17))x]surd(17)-10exp[xsurd(17)]cosh[1/2(5+surd(17))x]surd(17)-exp(xsurd(17))cosh[1/2(1+surd(17))x]surd(17)+34sinh[1/2(-5+surd(17))x]-34exp(xsurd(17))sinh[1/2(1+surd(17))x]+34exp(xsurd(17))cosh[1/2(5+surd(17))x]+17cosh[1/2(-1+surd(17))x]+cosh[1/2(-1+surd(17))x]surd(17)+17exp(xsurd(17)cosh[1/2(1+surd(17))x]+sinh[1/2(-1+surd(17))x]surd(17)+17sinh[1/2(-1+surd(17))x]+exp(xsurd(17))sinh[1/2(1+surd(17)x]surd(17)+10exp(xsurd(17)sinh[1/2(5+surd(17))x]surd(17)} exp[-1/2(-3+surd(17))x]

I dont know how to get this answer..... please help me to solve it... thanks!

http://viewmorepics.myspace.com/index.cfm?fuseaction=viewImage&friendID=2099216&albumID=0&imageID=7654718

please look it and tell me how to solve it....plz 3x!

the answer given not match with the answer that u gave.....before this...
So? how ? I have to pass up this question this friday...

http://i73.photobucket.com/albums/i215/aki_fashla/ODE.jpg

Answer 1

this is a non-HOMOGENEOUS ODE;
There are two ways to solve:
1st: Calculating. wroneskian
2nd: finding particular sol.
I prefer second way:
y=y(hom)+y(p)
homogeneous sol: y"-3y'-2y=0
this is a constant coefficient HOM.ODE, so the answer is in the kind of y1=Exp(m1x) & y1=Exp(m2x);
now we are going to find m1 & m2:
we write the characteristic eq. for this ODE:
m^2-3m-2=0; from here we have:
m1=1/2(3+sqrt(17)) & m2=1/2(3-sqrt(17))
the HOM. answer is any linear summation of these two answer say y(hom)=C1Exp(m1x)+C2Exp(m2x);
Now we are going to find Particular sol.
we consider particular sol in the form of y(p)=ASinh(x)+BCosh(x) (A&B are Const). we want to find A&B:
y(p) should satisfy the ODE So;
ASinh(x)+BCosh(x)-3(BSinh(x)+ACosh(x))-2(ASinh(x)+BCosh(x))=Sinh(x); Result:
setting sum of coefficients of Sinh(x) in the left side=1
setting sum of coefficients of Cosh(x) in the left side=0
A= 1/8
B= -3/8
y=C1Exp(m1x)+C2Exp(m2x)+ 1/8Sinh(x) - 3/8Cosh(x) where
m1=1/2(3+sqrt(17)) & m2=1/2(3-sqrt(17)) ||| C1&C2 arbitrary constants;
You can expand Sinh(x)=(exp(x)-exp(-x)) /2&
Cosh(x)=(exp(x)+exp(-x))/2
I hope this help;
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Answer 2

Didn't I already answer this question? Check this out:
http://img237.imageshack.us/img237/7020/2ndorderinhomogenefi3.png

EDITED 3/13/07:

It appears that you have a more extended particular solution. (I cannot see all the terms in your posting.) The complementary solution agrees with what I got.

The recommended form for the particular solution that I used came from
http://www.ucl.ac.uk/Mathematics/geomath/level2/deqn/de10.html#l1

It is possible that a more complex form is also a solution. I would need to see the work to identify the form that was used. Note that in the above article, after the table of proposed particular solutions, is this statement:

"You can see that it is actually fairly clear what you should try for the PI in each case, it's basically whatever f(x) is, but you do need to remember that the PI has other parts as well as just f(x)."

Perhaps there are "other parts".

NOTE: Your file on photobucket is too fuzzy to read. Try posting a larger size or at higher resolution.

Answer 3

its a second order DE. whats the problem.
why not just change sinh(x) to exponentials
sinh(x) = ( e^x - e^-x )/2