## Calculus 10th Edition

The particular solution that passes through the point $(3,4)$ is $$y=\sqrt{2x^2-2}.$$
Here the solution is given in its' implicit form $2x^2-y^2=C$ and it contains an arbitrary constant $C$. To find required $C$ we have to demand that the solution passes through the point $(3,4)$ i.e. when we put $x=3$ we get $y=4$: $$2\times3^2-4^2=C\Rightarrow C=2.$$ Now we have $$2x^2-y^2=2.$$ From here we will find $y$ explicitly: $$2x^2-y^2=2\Rightarrow y^2=2x^2-2\Rightarrow y=\pm\sqrt{2x^2-2}.$$ We have to throw away the solution with the $"-"$ sign because when we put $x=3$ we get $$y=-\sqrt{2\times3^2-2}=-\sqrt{16}=-4$$ which we did not want. When we take the $"+"$ sign we have for $x=3$ $$y=\sqrt{2\times3^2-2}=\sqrt{16}=4,$$ and that is exactly what we wanted. So the particular solution that passes through the point $(3,4)$ is $$y=\sqrt{2x^2-2}.$$