Question

x^2ydx+y^2xdy=0 , y(1)=1

Answer 1

Given:

$x^2ydx+y^2xdy=0,y(1)=1$

First, divide by xy:

$\begin{gathered} \begin{equation*} x^2ydx+y^2xdy=0 \end{equation*} \\ \frac{x^2ydx+y^2xdy}{xy}=0 \\ xdx+ydy=0 \\ ydy=-xdx \end{gathered}$

Then, integrate both sides:

$\begin{gathered} \int ydy=\int-xdx \\ \frac{y^2}{2}+C=-\frac{x^2}{2}+C \end{gathered}$

Simplify:

$y^2=C-x^2$

Now, since y(1) = 1, we will use this to solve for C to get a particular solution:

$\begin{gathered} y^{2}=C-x^{2} \\ (1)^2=C-(1)^2 \\ 1=C-1 \\ C=1+1 \\ C=2 \end{gathered}$

Therefore, our particular solution is:

$y^2=2-x^2$