What Shape Do the Characteristic Curves of This PDE Form?

[jimmycricket]

Homework Statement

suppose $$u(x,y)$$ satisfies the partial differential equation:
$$-4y\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0$$
Find the characteristic curves for this equation and name the shape they form

The Attempt at a Solution

$$\frac{dy}{dt}=1 \Longrightarrow y=t+y_0$$
$$\frac{dx}{dt}=-4y=-4(t+y_0) \Longrightarrow x=-2t^2-4y_0t+x_0$$

What can I say about the shape these curves form?

 

[pasmith]

Homework Statement

suppose $$u(x,y)$$ satisfies the partial differential equation:
$$-4y\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0$$
Find the characteristic curves for this equation and name the shape they form

The Attempt at a Solution

$$\frac{dy}{dt}=1 \Longrightarrow y=t+y_0$$
$$\frac{dx}{dt}=-4y=-4(t+y_0) \Longrightarrow x=-2t^2-4y_0t+x_0$$

What can I say about the shape these curves form?

$y$ is a first-order polynomial in $t$. $x$ is a quadratic in $t$. Therefore $x$ is a quadratic in $y$. What shape does the graph of a quadratic function have?