Find general solution of non homogenious linear equation

[Ortix]

Homework Statement

Find the general solution of the differential equation:
y'' + 4y' + 13y' = 169x + 81e^{e-2x}

EDIT: Can't get latex to work.. so :
http://mathbin.net/60293

The Attempt at a Solution

[EQ]13Ax + 4A + 13B + 9Ce^{-2x} = 169x+81e^{-2x}[/EQ]

http://mathbin.net/60294 Now i have no clue how to find A B C... I'm probably over thinking this :P

the answers are 13 -4 and 9...

EDIT2:
So i think i figured out how to get A and C. I divided 169 by 13 and 81 by 9, which makes sense. But what about B?

 

[Metaleer]

You need to identify coefficients on both sides of the equation, and thus arrive at a linear set of simultaneous equations in A, B and C.

 

[Ortix]

The same is said in the book, still doesn't get me any further...

 

[Metaleer]

OK, can you see that there's no number (independent term) on the RHS? This means that 4A + 13B must be 0. You already have A; solve for B.

 

[Ortix]

there we go :D so because of the fact that there is an x and a e^-2x on the RHS i can solve for Ax and the Ce^-2x since they have a corresponding term. And 4A + 13B = 0 because there are no terms (or whatever you want to call them) on the RHS. I think i understand it, i just can't put it into words :P

 

[Metaleer]

there we go :D so because of the fact that there is an x and a e^-2x on the RHS i can solve for Ax and the Ce^-2x since they have a corresponding term. And 4A + 13B = 0 because there are no terms (or whatever you want to call them) on the RHS. I think i understand it, i just can't put it into words :P

Indeed. Another way to get a set of equations in A, B and C would have been to give three different values to the x, but in this case, direct comparison works faster.

 

[epenguin]

For the particular integral of

y'' + 4y' + 13y' = 169x + 81e^{e-2x}

any linear combination of a particular integral of

y'' + 4y' + 13y' = 169x

and of a particular integral of

y'' + 4y' + 13y' = 81e^{e-2x}

will do, so you can deal with these separately if you want.

Personally I always forget the methods between times so a way I do things like this (equivalent to other methods) is to ask: the right hand side, call if f(x) - what differential equations does that satisfy? Or get f'(x) and f"(x) and ask how can you make your original d.e. using these and f(x)? Or put it like this: if y is an n degree polynomial, (ay" + by' + cy) is an n-degree polynomial, if y is an exponential (ay" + by' + cy) is an exponential, if y is a combination of sines and cosines, (ay" + by' + cy) is a combination of sines and cosines so get f'(x) and f"(x) as well as f, and you can expect to be able to fit these assumed forms of solution if you hammer their constants to fit. For other forms of f this may not be possible or not so straightforward, but these three cases cover a lot of the elementary physical applications.