Very simple diff eq [concept help needed for exam]

[Tom McCurdy]

Homework Statement

Find the exact solution to this problem
y'=4x–y+9;y(0)=6

The Attempt at a Solution

I am panicing because I have an exam tomorrow and I can't remeber a lot of the basics for diff eq. I tried to solve this using 2nd order style...

first I made it

y'-y=4x+9

For natural solution i do
r+1=0
r=-1

natural solution = A*e^(-x)

How do I get the forced soution?


And how would I do it on a second order equation so like

y''+By'+y = some x terms

I know how to do natural solutions... its the forced that keeps getting me

 

[d_leet]

You do realize that that is a first order differential equation right? Do you know the standard method of integrating factors for finding the solution of a first order differential equation?

 

[Tom McCurdy]

I am trying to avoid first order methods, I would like to be able to solve it using second order method... since r is a real root it would take the natural form A*e(-rx)

Really I guess I picked a bad example... I thought it would be easier than second order... I am trying to figure out how to solve for a second order "forced solution"

 

[d_leet]

Why are you trying to avoid first order methods for a first order equation? It really is easiest to just find an integrating factor and solve the equation that way.

 

[Tom McCurdy]

I know its the easiest way, but I don't have any first order problems on my upcomming exam, I just thought it would be easier to learn how to do forced solutions on a first order problem (which I don't even know if it is possilbe) I know how to do the integrating factor method.

 

[SGT]

If your forcing function is a polynomial, the particular solution will be a polynomial of the same degree. In your case
$$y_p = Ax + B$$
Substitute $$y_p$$ and its derivative in the differential equation to obtain the parameters A and B.