To show solutions of a linear system lie on parabolas in phase space.

Question

Given a linear system

$\dot{x}=x$

$\dot{y}=2y$

To show solutions of a linear system lie on parabolas in phase space. Which solutions (if any) do not lie on parabolas?

It is the second question that bothers me. I have solved the system:

$\frac{x}{x_0}=(\frac{y}{y_0})^{1/2}$ which is a parabola I suppose.

Answer 1

Your equation $$ \frac{x}{x_0} = \left(\frac{y}{y_0}\right)^{1/2} $$ won't hold if one (or both) of $x_0$ or $y_0$ is zero.

For instance, if $x_0 = 0$, then $x(t) = 0$ for all $t$. Meanwhile $y(t) = y_0 e^{2t}$. The solution curve lies in the $y$-axis.

On the other hand, if $y_0 = 0$, then $y(t) = 0$ for all $t$. Meanwhile $x(t) = x_0 e^t$. The solution curve lies in the $x$-axis.

You might think of these curves as the limits of the curves $x y_0^2 = x_0 y^2$ as $x_0 \to 0$ or $y_0 \to 0$.