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I have the following DE:

$$ xy' = y + x\cos^2\left(\frac{y}{x}\right) $$

I then rule out the possible methods of solving it:

  • Not separable
  • Not homogeneous
  • Not exact
  • Possible integrating factor in $x$? No.
  • Possible integrating factor in $y$? No.
  • Not linear

Above are the only ways I have learned to solve DEs. To help, I've rewritten the DE in this form:

$$ M(x,y)dx + N(x,y)dy = 0 $$ $$ \left(y + x\cos^2\left(\frac{y}{x}\right)\right)dx - xdy =0 $$ $$ M_{y} = 1 - 2\cos\left(\frac{y}{x}\right)\sin\left(\frac{y}{x}\right) $$ $$ N_{x} = -1 $$

I'm completely lost now. I can't seem to find any integrating factors (the $\frac{y}{x}$) inside the trigs are making it so that I can't get things only in terms of $x$ or $y$.

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    Instead of «can't seem» in your title you probably mean «don't seem»... In any case, it is usually best to be generous to the non native English speakers and write out things :)2012-01-20
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    Why do you say the equation is not homogeneous?2012-01-20
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    Hint: rewrite this as a differential equation involving the function $z:x\mapsto y(x)/x$. (+1 for showing what you tried.)2012-01-20
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    @MarianoSuárez-Alvarez: Maybe I misunderstand how to count the powers. In the $M(x,y)$ and $N(x,y)$ form of the equation, I see the power of $y$ to be 1, the power of $xcos^2(y/x)$ to be 3 and the degree of $-x$ to be 1. Is that wrong?2012-01-20
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    @DidierPiau, I'm not familiar with that form and I'd rather solve it in a form I have learned in class :) I don't want to get mixed up.2012-01-20
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    I don't understand how you are "couting powers" of $x$ or $y$ in an expression such as $x\cos^2(y/x)$ (and I am pretty sure that that does not mean anything, really), but your equation *is* homogeneous (Paul's answer is using exactly this fact)2012-01-20
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    @MarianoSuárez-Alvarez: Yeah I've just realized his example is using $u$, which we use when we deal with homogeneous polynomials. I'm going to need to get a better definition of homogeneous :)2012-01-20
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    A useful definition of homogeneous is: rewrite the equation in the form $y'=F(x,y)$ for some function $F$. Then the equation is homogeneous if for all (non-zero...) scalars $\lambda$ we have $F(\lambda x,\lambda y)=F(x,y)$.2012-01-20
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    @MaxMackie: Where is the *mix up*?2012-01-20
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    x(xdy-ydx/x^2)=cos^2(y/x)dx x d(-y/x)=1/sec^2(y/x)dx sec^2(y/x) d(-y/x)=1/x dx sec^2(y/x)=(1/x/y/x)dx integration sec^2(y/x)dx=(1/y)dx tan(y/x)=logy2012-10-07
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    This is the right answer unless I made a mistake in my calculations. $$y=x\arctan(kln(x))$$ k=const.2012-10-07

1 Answers 1

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Dividing both sides by $x$, we get $$\tag{1}y' =\frac{y}{x} + \cos^2\left(\frac{y}{x}\right).$$ If we let $u=\displaystyle\frac{y}{x}$, then we have $y=ux$ and $$\frac{dy}{dx}=x\frac{du}{dx}+u.$$ Substitute this into $(1)$, we have $$x\frac{du}{dx}+u=u+\cos^2(u).$$ I think you can solve it starting from here.