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anonymous
 one year ago
Hello all, I'm having a hard time with this:
y'y/2x=xsin(x/y) ... it seems an homogeneus with z=x/y sostitutions but it don't works. Any idea?
anonymous
 one year ago
Hello all, I'm having a hard time with this: y'y/2x=xsin(x/y) ... it seems an homogeneus with z=x/y sostitutions but it don't works. Any idea?

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[y'\frac{y}{2x}=x\sin\frac{x}{y}~~?\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0There's a nice general way to check if a given ODE is homogeneous. Given an ODE of the form \(y'=f(x,y)\), the ODE is homogeneous if for some real \(\alpha\), \(f(t x,t y)=t^\alpha f(x,y)\). Let's check: \[y'\frac{y}{2x}=x\sin\frac{x}{y}~~\implies~~f(x,y)=\frac{y}{2x}+x\sin\frac{x}{y}\] So you have \[\begin{align*} f(tx,ty)&=\frac{ty}{2tx}+tx\sin\frac{tx}{ty}\\[1ex] &=\frac{y}{2x}+tx\sin\frac{x}{y}\\[1ex] &\neq t^\alpha f(x,y)&\text{for any }\alpha\in\mathbb{R} \end{align*}\] and so the substitution you mentioned would not work.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Actually, your substitution isn't the one one usually uses for homogeneous equations. I think you meant \(y=zx\) i.e. \(z=\dfrac{y}{x}\)?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Thank you for your checking method! I've tryed both \[y/x \] and that unhortodox \[x/y \] for the substitution but nothing. Can we say we have not elementary solution for this ODE?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I would say so (though that doesn't mean I'm not wrong). Like IrishBoy mentioned, the \(\sin\) term is the main source of trouble. Had its argument been just \(x\) instead of \(\dfrac{x}{y}\), there would have been a chance. Numerical methods may be the best bet, but even then you would need an initial condition which you don't seem to have.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0No, I havent initial contition, this is the text "nude and crude" ... I'm suspecting there is a mistake in the text even if it is an official exam at my University. I will make some investigations again and let you know :). Thank you by now for you time (And sorry for my non perfect English)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0It's sure: was a mistake in the text :)
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