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How to solve the following DE $$ {dx \over x^2 - y^2 - z^2} = {dy \over 2xy} = {dz \over 2xz}$$ Equating the last part, I got $y = c_1 z$ and then I'm stuck. substituting the value of $y$ and equating first and last gives $$ {dx \over dz} = {x^2 - z^2(c_1^2 + 1) \over 2xz}$$ How to solve it?

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Have you learned about homogeneous forms of ODEs? See if there's some single variable in terms of $x,y$ in which you could write the expression $\frac{dx}{dz}$. – Eugene Shvarts Aug 2 '12 at 9:56
up vote 1 down vote accepted

It is homogeneous in $x$ and $z$; that is, dividing top and bottom of the fraction by $z^2$ on the RHS gives

$$\dfrac{dx}{dz} = \dfrac{X^2 - (c_1^2+1)}{2X}$$

where $X = \dfrac{x}{z}$. So substitute $x=zX$ into the equation and you'll be able to solve it.

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Aaand don't forget to differentiate the relation $x = zX$ to recover the appropriate relation for the new variable's derivative. – Eugene Shvarts Aug 2 '12 at 10:00

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