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use reduction of order to solve the problem...
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x''(t)-4x(t)=2. First find the homogeneous x''(t)-4x(t)=0, characteristic equation m^2-4=0, m1/2=+-2, xh=C1*e^(m1*t)+C2*e^(m2*t)=C1e^(2*t)+C2*e^(-2*t). As we see that a non homogeneous part in start equation is 2, xp must be constant. x=xh+xp. Substitute this in start equation (C1*e^2t+C2*e(-2t)+xp)''-4(C1*e^2t+C2*e(-2t)+xp)=2, xp''=0, -4xp=2, xp=-1/2.
Kind regards,
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derdack, yes, the problem can be solved that way, but I don't believe it is a god idea to advise a person to ignore the teacher's or text's instructions. Slapmaxwell1 says he was instructed to use "reduction of order". It is not at all uncommon to ask a person to solve a simple problem using a more complicated technique in order to practice that technique.
Slapmaxwell1, yes, letting $\displaystyle y= ue^{2x}$ reduces the left side of the equation to $\displaystyle e^{-2x}(u''- 4u')$ but you appear to have left out the right side of the equation. You have $\displaystyle e^{-2x}(u''- 4u')= 2$ so that $\displaystyle u''- 4u'= 2e^{-2x}$ and, letting w= u', $\displaystyle w'- 4w= 2e^{-2x}$.
That is now a first order equation linear equation. It has $\displaystyle e^{-4x}$ as integrating factor:
$\displaystyle e^{-4x}w'- 4e^{-4x}w= (e^{-4x}w)'= 2e^{-4x}$
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You are right, but it is completly corect that we have other ways for solving which is important for learner. (John - Beautiful mind) .
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Quote:
Originally Posted by
derdack 
You are right, but it is completly corect that we have other ways for solving which is important for learner. (John - Beautiful mind) .
True, but I was taught reduction of order before your method. The different methods come in their own itme.
-Dan
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Yes. Both of us are right...
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i am starting to see why you guys were so pissed that i wasnt using latex...I apologize. learning the commands in beginning is a bit troublesome and take a lil extra time, but its totally worth the effort.
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ok ok i see it now.
if i let $\displaystyle y_{p}=A$ then $\displaystyle y'=0 , y''=0$
ok so $\displaystyle 0-4A=2$ so $\displaystyle A=\frac{-1}{2}$
there for $\displaystyle y_{p}=\frac{-1}{2}$ so $\displaystyle y=y_{c}+y_{p}$ because i found Yc I would just put it all together for my final answer. thanks again!
