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ZachN
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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.
(x+y){dx} - (x-y){dy} = 0
y=ux, {dy} = u{dx} + x{du}
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.
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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