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*} \\ (x^2ydx+y^2xdy)/(xy)=0 \\ xdx+ydy=0 \\ ydy=-xdx \end{gathered}`

Then, integrate both sides:

`\begin{gathered} \int ydy=\int-xdx \\ (y^2)/(2)+C=-(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`