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?