Equation w/ Homogeneous Coefficients - y=ux substitution

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Differential Equation w/ Homogeneous Coefficients - y=ux substitution

I am teaching myself, this problem is from ODEs by Tenenbaum and Pollard. This is not homework for a class.

Homework Statement



(x+y){dx} - (x-y){dy} = 0

Homework Equations



y=ux, {dy} = u{dx} + x{du}

The Attempt at a Solution



Substitution should lead to a separable equation in x and u:

(x+ux){dx} + (ux - x)(u{dx} + x{du}) = 0;
x(u+1)dx + u2x{dx} + ux2{du} - ux{dx} - x2{du} = 0;
xu(2){dx} + x2(u - 1){du} = 0;
-1/x{dx} = (u - 1)/(u2 + 1){du}

Okay, I am assuming that I am correct up to this point but the answer given by the text is:

Arc tan(y/x) - 1/2log(x2 + y2) = c

I understand where the Arc tan(y/x) comes from - the (1/(u2 + 1)). I am having trouble with the 1/2 log(x2 + y2) - where does the -log(x) go?

I have a couple of other problems in the same form as this which arise in isogonal trajectories and I don't want to just skip over this because I am obviously having a problem with these integrations.
 
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Answers and Replies

  • #2
HallsofIvy
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I am teaching myself, this problem is from ODEs by Tenenbaum and Pollard. This is not homework for a class.

Homework Statement



(x+y){dx} - (x-y){dy} = 0

Homework Equations



y=ux, {dy} = u{dx} + x{du}

The Attempt at a Solution



Substitution should lead to a separable equation in x and u:

(x+ux){dx} + (ux - x)(u{dx} + x{du}) = 0;
x(u+1)dx + u2x{dx} + ux2{du} - ux{dx} - x2{du} = 0;
xu(2){dx} + x2(u - 1){du} = 0;
-1/x{dx} = (u - 1)/(u2 + 1){du}
so -1/x dx= u/(u2+ 1)du - 1/(u2+1 du
The left side give -ln(x), of course. Then anti-derivative of -1/(u2+ 1) is -arctan(u) but to integrate u/(u2+ 1) let v= u2+ 1, dv= 2udu and the integral becomes udu/(u2+ 1)= dv/(2v)= (1/2)ln(v)= (1/2) ln(u2+ 1)= (1/2)ln((y2/x2+ 1)= (1/2)ln((x2+ y2)/x2= (1/2)ln(x2+ y2)- ln(x).

Okay, I am assuming that I am correct up to this point but the answer given by the text is:

Arc tan(y/x) - 1/2log(x2 + y2) = c

I understand where the Arc tan(y/x) comes from - the (1/(u2 + 1)). I am having trouble with the 1/2 log(x2 + y2) - where does the -log(x) go?

I have a couple of other problems in the same form as this which arise in isogonal trajectories and I don't want to just skip over this because I am obviously having a problem with these integrations.
 
  • #3
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Yes, thank you - I was not splitting up the ration into two equations and then integrating. I will try to be more observant from now on.
 
  • #4
tiny-tim
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Homework Helper
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(x+y){dx} - (x-y){dy} = 0

the answer given by the text is:

Arc tan(y/x) - 1/2log(x2 + y2) = c

Hi ZachN! :smile:

As an alternative method … always look at the answer … it may give you a clue as to an easy substitution …

in this case, the answer uses tan-1y/x and x2 + y2, so the obvious substitution is into polar coordinates, r and θ.

Try it and see. :smile:
 
  • #5
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I'll try polar coordinates.
 

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