Given:
[tex]x^2ydx+y^2xdy=0,y(1)=1[/tex]First, divide by xy:
[tex]\begin{gathered} \begin{equation*} x^2ydx+y^2xdy=0 \end{equation*} \\ \frac{x^2ydx+y^2xdy}{xy}=0 \\ xdx+ydy=0 \\ ydy=-xdx \end{gathered}[/tex]Then, integrate both sides:
[tex]\begin{gathered} \int ydy=\int-xdx \\ \frac{y^2}{2}+C=-\frac{x^2}{2}+C \end{gathered}[/tex]Simplify:
[tex]y^2=C-x^2[/tex]Now, since y(1) = 1, we will use this to solve for C to get a particular solution:
[tex]\begin{gathered} y^{2}=C-x^{2} \\ (1)^2=C-(1)^2 \\ 1=C-1 \\ C=1+1 \\ C=2 \end{gathered}[/tex]Therefore, our particular solution is:
[tex]y^2=2-x^2[/tex]