A tricky 2nd Oder ODE Problem,nonhomo delta=0 case

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solve the initial value problem:
y''+8.4y'+17.64y=e-4.2x
y(0)=1, y'(0)=1

y(x)=?

The way I tried to achieve is to solve the corresponding homo equation first:
y''+8.4y'+17.64y=0, which gives yc;
yc=c1e-4.2x+c2e-4.2xx

Then try to find yp, generally I would assume a yp=Ae-4.2x, but from the yc got above, clearly yp=Ae-4.2x or yp=Ae-4.2xx is not good. If I add one more 'x' in yp assumption,in which it seems A has to be 0, which is not right either....

Any idea? Or I made some mistake?

Meric.
 

Answers and Replies

  • #2
HallsofIvy
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Because both [itex]e^{-4.2x}[/itex] and [itex]xe^{-4.2x}[/itex] satisfy the homogeneous equation, you need "go up" one more x: try
[tex]y_p= x^2e^{-4.2x}[/tex]

Was that what you meant by "add one more 'x' in yp assumption"?

If
[tex]y_p= Ax^2e^{-4.2x}[/tex]
then
[tex]y_p'= 2Axe^{-4.2x}- 4.2Ax^2e^{-4.2x}[/tex]
and
[tex]y_p"= 2Ae^{-4.2x}- 16.8Axe^{-4.2x}+ 17.64Ax^2e^{-4.2x}[/tex]
Putting that into the differential equation, all the terms involving [itex]xe^{-4.2x}[/itex] or [itex]x^2e^{-4.2x}[/itex] will cancel leaving only [itex]2Ae^{-4.2}= e^{-4.2x}[/itex] so A is NOT 0.
 
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  • #3
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Oh I made a silly mistake in differentiate the Yp...

Yeah, you are right, I got the right answer now. Thanks
 

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