Final Answer
$y=\sqrt{\left(-\sqrt{x^2-2}+C_0\right)^{2}+2},\:y=-\sqrt{\left(-\sqrt{x^2-2}+C_0\right)^{2}+2}$
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Step-by-step Solution
Specify the solving method
1
Grouping the terms of the differential equation
$y\sqrt{x^2-2}dy=0-x\sqrt{y^2-2}dx$
2
$x+0=x$, where $x$ is any expression
$y\sqrt{x^2-2}dy=-x\sqrt{y^2-2}dx$
3
Group the terms of the differential equation. Move the terms of the $y$ variable to the left side, and the terms of the $x$ variable to the right side of the equality
$\frac{y}{\sqrt{y^2-2}}dy=\frac{-x}{\sqrt{x^2-2}}dx$
4
Integrate both sides of the differential equation, the left side with respect to
$\int\frac{y}{\sqrt{y^2-2}}dy=\int\frac{-x}{\sqrt{x^2-2}}dx$
5
Take out the constant $-1$ from the integral
$\int\frac{y}{\sqrt{y^2-2}}dy=-\int\frac{x}{\sqrt{x^2-2}}dx$
6
Solve the integral $\int\frac{y}{\sqrt{y^2-2}}dy$ and replace the result in the differential equation
$\sqrt{y^2-2}=-\int\frac{x}{\sqrt{x^2-2}}dx$
7
Solve the integral $-\int\frac{x}{\sqrt{x^2-2}}dx$ and replace the result in the differential equation
$\sqrt{y^2-2}=-\sqrt{x^2-2}+C_0$
8
Find the explicit solution to the differential equation. We need to isolate the variable $y$
$y=\sqrt{\left(-\sqrt{x^2-2}+C_0\right)^{2}+2},\:y=-\sqrt{\left(-\sqrt{x^2-2}+C_0\right)^{2}+2}$
Final Answer
$y=\sqrt{\left(-\sqrt{x^2-2}+C_0\right)^{2}+2},\:y=-\sqrt{\left(-\sqrt{x^2-2}+C_0\right)^{2}+2}$