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Idealist10
 one year ago
Find the general solution of (x2)(x1)y'(4x3)y=(x2)^3.
Idealist10
 one year ago
Find the general solution of (x2)(x1)y'(4x3)y=(x2)^3.

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myininaya
 one year ago
Best ResponseYou've already chosen the best response.1so i assumed you tried to put in the form y'+p*y=q form

Idealist10
 one year ago
Best ResponseYou've already chosen the best response.1Please help me. How do I start? Divide (x2)?

myininaya
 one year ago
Best ResponseYou've already chosen the best response.1divide by (x2)(x1)

Idealist10
 one year ago
Best ResponseYou've already chosen the best response.1Okay, I'll try and tell me what's going on. Please don't leave.

myininaya
 one year ago
Best ResponseYou've already chosen the best response.1you may have to some partial fractions just to let you know

Idealist10
 one year ago
Best ResponseYou've already chosen the best response.1But how do I integrate (4x3)/((x2)(x1))?

Idealist10
 one year ago
Best ResponseYou've already chosen the best response.1So do I multiply (x2)(x1)? It's x^23x+2.

myininaya
 one year ago
Best ResponseYou've already chosen the best response.1\[\frac{34x}{(x2)(x1) }=\frac{A}{x2}+\frac{B}{x1}\] 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
 one year ago
Best ResponseYou've already chosen the best response.1So 4x/(x2)3/(x1)? And 4x(x1)3(x2)? But equal to what?

myininaya
 one year ago
Best ResponseYou've already chosen the best response.1... Have you ever done partial fractions before?

Idealist10
 one year ago
Best ResponseYou've already chosen the best response.1So isn't it 5ln(x2)ln(x1)?

myininaya
 one year ago
Best ResponseYou've already chosen the best response.1I will show an example: \[\frac{1}{(x5)(x3)}\] 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/(x5)+B/(x3) To find what A and B are we will need to combine those fractions in put in to form that is 1/[(x5)(x3)] form. so let's do that \[\frac{A}{x5}+\frac{B}{x3}=\frac{A(x3)+B(x5)}{(x5)(x3)}=\frac{(A+B)x3A5B}{(x5)(x3)}\] but the other side is equal to 1/[(x5)(x3)] 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 3A5B=1 so we have a system of linear equations to solve: A+B=0 3A5B=1  To solve I will try to setup for elimination Multiply first equation by 3 3A+3B=0 3A5B=1 now add 2B=1 B=1/2 If B=1/2 and A+B=0, then A=1/2 So you know that \[\frac{1}{(x5)(x3)} \text{ can be written as } \frac{1}{2} \frac{1}{x5}\frac{1}{2}\frac{1}{x3}\] and we know how to intgrate both of those terms

myininaya
 one year ago
Best ResponseYou've already chosen the best response.1@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 \[v=e^{5\lnx2+\lnx1}\] This can be a lot prettier

myininaya
 one year ago
Best ResponseYou've already chosen the best response.1Now 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
 one year ago
Best ResponseYou've already chosen the best response.1What I'm saying is the section on partial fractions can not be explained by one example

myininaya
 one year ago
Best ResponseYou've already chosen the best response.1Anyways back to what you were saying...

myininaya
 one year ago
Best ResponseYou've already chosen the best response.1to write your integrating factor a lot prettier you will need to recall some law of exponents and also some log properties

myininaya
 one year ago
Best ResponseYou've already chosen the best response.1you do recall that x^(a+b) can be written as (x^a)(x^b) ?

myininaya
 one year ago
Best ResponseYou've already chosen the best response.1do you see how that could be helpful here?

Idealist10
 one year ago
Best ResponseYou've already chosen the best response.1Yes, wait a second.

Idealist10
 one year ago
Best ResponseYou've already chosen the best response.1So I got v=(x1)/(x2)^5, is this right?

Idealist10
 one year ago
Best ResponseYou've already chosen the best response.1Let me work it out then.

Idealist10
 one year ago
Best ResponseYou've already chosen the best response.1Yes, I got it! Thank you so much! Wow, this is a tough problem to solve!

myininaya
 one year ago
Best ResponseYou've already chosen the best response.1Probably mostly because of the partial fractions part?

myininaya
 one year ago
Best ResponseYou've already chosen the best response.1I 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
 one year ago
Best ResponseYou've already chosen the best response.1just practice practice it will sink in eventually
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