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Math Help - solve the differential...

  1. #1
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    solve the differential...

    ok so i did this problem and checked in the back of my book and got it wrong..im hoping someone can show me the error of my work.

    (y^2+yx)dx - x^2dy = 0

    ok so there are a couple techniques that i can use to solve the problem...

    1. almost exact
    2. homogeneous

    im going with homogeneous, because i think its a little easier to use and would require less steps.

    so i make my substitution y= ux
    therefore dy = udx +xdu

    then after i make my substitution..

    (u^2x^2 + ux^2) dx - x^2(udx + xdu)

    after doing some algebra i got a final answer of

    y = xC1 - x/ln|x|

    C1 = 1/C

    originally i got x/y = C - ln|x| i just flipped it and redefined my C, not sure if i can do this but it felt good? i dunno..lol thanks in advance.
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  2. #2
    A Plied Mathematician
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    after doing some algebra i got a final answer of

    y = xC1 - x/ln|x|

    C1 = 1/C

    originally i got x/y = C - ln|x| i just flipped it and redefined my C, not sure if i can do this but it felt good? i dunno..lol thanks in advance.
    Yeah, I think your problem is in here somewhere. x/y = C - ln|x| is correct. Be more careful with your algebra: subtraction and division do not commute!
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  3. #3
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by slapmaxwell1 View Post
    ok so i did this problem and checked in the back of my book and got it wrong..im hoping someone can show me the error of my work.

    (y^2+yx)dx - x^2dy = 0

    ok so there are a couple techniques that i can use to solve the problem...

    1. almost exact
    2. homogeneous

    im going with homogeneous, because i think its a little easier to use and would require less steps.

    so i make my substitution y= ux
    therefore dy = udx +xdu

    then after i make my substitution..

    (u^2x^2 + ux^2) dx - x^2(udx + xdu)

    after doing some algebra i got a final answer of

    y = xC1 - x/ln|x|

    C1 = 1/C

    originally i got x/y = C - ln|x| i just flipped it and redefined my C, not sure if i can do this but it felt good? i dunno..lol thanks in advance.
    OK, from here (u^2x^2+ux^2)dx-x^2(udx+xdu)=0 we see that

    x^2u^2dx-x^3du=0 and after dividing by u^2x^3 we integrate: \int\frac{dx}{x}-\int\frac{du}{u^2}=0

    which means that

    \ln|x|+\frac{1}{u}=c. Now back sub.
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  4. #4
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    ok im gonna rework the problem..i was hoping the book might have been wrong..one more quick question..i know i did the problem as homogeneous, but could i have done it as an almost exact differential? find a substitution that would make it exact and get my answer that way? thanks in advance.
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  5. #5
    A Plied Mathematician
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    Quote Originally Posted by slapmaxwell1 View Post
    ok im gonna rework the problem..i was hoping the book might have been wrong..one more quick question..i know i did the problem as homogeneous, but could i have done it as an almost exact differential? find a substitution that would make it exact and get my answer that way? thanks in advance.
    Possibly. I'm not very familiar with that technique, although I do know some DE's succumb to it.
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