Idealist10
  • Idealist10
Find the general solution of (x-2)(x-1)y'-(4x-3)y=(x-2)^3.
Mathematics
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SOLVED
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katieb
  • katieb
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Idealist10
  • Idealist10
@myininaya
Idealist10
  • Idealist10
@beccaboo333
myininaya
  • myininaya
so i assumed you tried to put in the form y'+p*y=q form

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Idealist10
  • Idealist10
Please help me. How do I start? Divide (x-2)?
myininaya
  • myininaya
divide by (x-2)(x-1)
Idealist10
  • Idealist10
Okay, I'll try and tell me what's going on. Please don't leave.
myininaya
  • myininaya
you may have to some partial fractions just to let you know
Idealist10
  • Idealist10
But how do I integrate (4x-3)/((x-2)(x-1))?
myininaya
  • myininaya
partial fractions
Idealist10
  • Idealist10
So do I multiply (x-2)(x-1)? It's x^2-3x+2.
myininaya
  • myininaya
no
myininaya
  • myininaya
\[\frac{3-4x}{(x-2)(x-1) }=\frac{A}{x-2}+\frac{B}{x-1}\] before you integrate you write what is on the left hand side in the right hand sides's form you must find A and B to complete that transaction
Idealist10
  • Idealist10
So 4x/(x-2)-3/(x-1)? And 4x(x-1)-3(x-2)? But equal to what?
myininaya
  • myininaya
... Have you ever done partial fractions before?
Idealist10
  • Idealist10
So isn't it 5ln(x-2)-ln(x-1)?
myininaya
  • myininaya
I will show an example: \[\frac{1}{(x-5)(x-3)}\] pretend we wanted to write this as a sum of proper fractions we can attempt to do this by trying to write that fraction in this form A/(x-5)+B/(x-3) To find what A and B are we will need to combine those fractions in put in to form that is 1/[(x-5)(x-3)] form. so let's do that \[\frac{A}{x-5}+\frac{B}{x-3}=\frac{A(x-3)+B(x-5)}{(x-5)(x-3)}=\frac{(A+B)x-3A-5B}{(x-5)(x-3)}\] but the other side is equal to 1/[(x-5)(x-3)] so that means the other can have no x's so A+B would have to be 0 and the constant term will have to be equal to 1 therefore -3A-5B=1 so we have a system of linear equations to solve: A+B=0 -3A-5B=1 ----------- To solve I will try to setup for elimination Multiply first equation by 3 3A+3B=0 -3A-5B=1 -----------now add -2B=1 B=-1/2 If B=-1/2 and A+B=0, then A=1/2 So you know that \[\frac{1}{(x-5)(x-3)} \text{ can be written as } \frac{1}{2} \frac{1}{x-5}-\frac{1}{2}\frac{1}{x-3}\] and we know how to intgrate both of those terms
myininaya
  • myininaya
@Idealist10 for the integrating factor there was a negative sign in front of y term so take what you have and multiply it by -1 but don't forget the integrate factor has that base e thing so for integrate factor we should have \[v=e^{-5\ln|x-2|+\ln|x-1|}\] This can be a lot prettier
myininaya
  • myininaya
Now that was just one example above from partial fractions There are other things to consider when writing a fraction composed of polynomials as a sum of fractions
myininaya
  • myininaya
What I'm saying is the section on partial fractions can not be explained by one example
myininaya
  • myininaya
Anyways back to what you were saying...
myininaya
  • myininaya
to write your integrating factor a lot prettier you will need to recall some law of exponents and also some log properties
myininaya
  • myininaya
you do recall that x^(a+b) can be written as (x^a)(x^b) ?
Idealist10
  • Idealist10
Yes.
myininaya
  • myininaya
do you see how that could be helpful here?
Idealist10
  • Idealist10
Yes, wait a second.
Idealist10
  • Idealist10
So I got v=(x-1)/(x-2)^5, is this right?
myininaya
  • myininaya
Very good!
Idealist10
  • Idealist10
Let me work it out then.
Idealist10
  • Idealist10
Y
Idealist10
  • Idealist10
Yes, I got it! Thank you so much! Wow, this is a tough problem to solve!
myininaya
  • myininaya
Probably mostly because of the partial fractions part?
Idealist10
  • Idealist10
Yes!
myininaya
  • myininaya
I guess you are in differential equations and try to recall all the things you learned in calculus 2. I do agree I had some trouble but with practice it should become easier on what you should do when you see something like that.
myininaya
  • myininaya
just practice practice it will sink in eventually

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