- #1

CptXray

- 23

- 3

## Homework Statement

For a vector field $$\begin{equation}

X:=y\frac{\partial{}}{\partial{x}} + x\frac{\partial{}}{\partial{y}}

\end{equation}$$

Find it's integral curves and the curve that intersects point $$p = \left(1, 0 \right).$$

Show that $$X(x,y)$$ is tangent to the family of curves: $$x^2 - y^2 = k,k∈ℝ$$

## Homework Equations

## The Attempt at a Solution

I know that a integral curve here is:

$$

\begin{bmatrix}

\dot{x} \\

\dot{y}\\

\end{bmatrix}

=

\begin{bmatrix}

0 & 1 \\

1 & 0\\

\end{bmatrix}

\begin{bmatrix}

x(t)\\

y(t)\\

\end{bmatrix}$$

Solving these gives me:

$$

\begin{cases}

x(t) = yt + x_{0}

& \\

y(t) = xt + y{0}

\end{cases}

$$

For point (1, 0):

$$

\begin{cases}

x(0) = 0 \rightarrow x_{0} = 1

& \\

y(0) = 0 \rightarrow y_{0} = 0

\end{cases}

$$

I guess that's what I was supposed to do here but i can't find a way to prove that $$x^2 - y^2 = k

$$

I'd be glad for help because I couldn't find anything helpful in my textbooks.

P.S.

Hello people, I'm new and happy to find this place :)