Inhomogeneous 2nd ODE

  • Thread starter Tzabcan
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Main Question or Discussion Point

If i have

3y" - 2y' -y = 14 + e2x+8x

And i want to find the general solution.

Obviously first i obtain the characteristic eqn, yc, by making it into a homogeneous eqn. Then i can get yp

BUT

Am i able to get yp for the e2x and the 14 + 8x separately, then add them together for yp?


Thanks
 

Answers and Replies

  • #2
pasmith
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If i have

3y" - 2y' -y = 14 + e2x+8x

And i want to find the general solution.

Obviously first i obtain the characteristic eqn, yc, by making it into a homogeneous eqn. Then i can get yp

BUT

Am i able to get yp for the e2x and the 14 + 8x separately, then add them together for yp?
Yes, because the ODE is linear.
 
  • #3
Geofleur
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I would like to expand a little on what pasmith said. Suppose that we have ## y'' - 2y' - y = f(x) + g(x) ## where ## f ## and ## g ## are some functions, and further that we have found solutions ## y_{p1} ## and ## y_{p2} ## such that

## y''_{p1} - 2y'_{p1} - y_{p1} = f(x) ## and

## y''_{p2} - 2y'_{p2} - y_{p2} = g(x) ##.

Then, if we define ## y_p = y_{p1} + y_{p2} ##, we will have

## y''_p - 2y'_p - y_p = (y_{p1}+y_{p2})'' - 2(y_{p1}+y_{p2})' - (y_{p1}+y_{p2}) = y''_{p1} + y''_{p2} - 2y'_{p1} - 2y'_{p2} - y_{p1} - y_{p2} = f(x) + g(x) ##.

If we had somewhere in the differential equation a term like, say, ## y^2 ##, the trick above would no longer work (I recommend trying it to see why).
 
  • #4
HallsofIvy
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An obvious "try" would be [itex]y= Ax+ B+ Ce^{2x}[/itex]. Put that into the equation and solve for A, B, and C.
 

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