Invariant: Differential Equations. I came across a question.
Show:
$A = x^2 + y^2 $
and
$B=x^2+z^2$ .    
Are invariants of the nonlinear system:
$x'(t)=y(t)z(t) .    
y'(t) = -x(t)z(t) .   
z'(t) = -x(t)y(t) . 
$
Now I know the solution is:
$A′ = 2xx′ + 2yy′ = 2xyz − 2xyz = 0$ .
and
$B′ = 2xx′ + 2zz′ = 0.$ 

My question is:
  a) What is the significance of invariant solutions?
  b) Is there anywhere in ordinary differential equations that invariant solutions become useful?
  c) Does invariance have any influence on existence theory?

 A: Invariant quantities are ridiculously useful in solving differential equations!  For example, if an asteroid is moving around the sun, it will move in a plane.  We can describe its position in terms of a distance $r(t)$ from the sun and an angle $\theta(t)$ as it goes around the sun.  The differential equations that govern the motion of this asteroid are
$$
r'' = - \frac{\gamma}{r^2} + r (\theta')^2
$$
and
$$
\theta'' + 2 r' \theta' = 0.
$$
These are coupled, non-linear ODEs, and you'd have a hard time if you ever wanted to write down a solution directly from these equations.
But the system has two invariants that make things much easier.  First, the second equation implies that $r^2 \theta'$ is an invariant of the system;  it's (proportional to) the asteroid's angular momentum.  If we call this constant $\ell$, then we have $\theta' = \ell/r^2$, which means that our first equation becomes
$$
r'' = - \frac{\gamma}{r^2} + r \left( \frac{\ell}{r^2} \right)^2 = - \frac{\gamma}{r^2} + \frac{\ell^2}{r^3}.
$$
Already this is a simplification;  we've gone from a set of two couple second-order differential equations to a single second-order differential equation.  What's more, you can show that another invariant of the system is the quantity
$$
e = \frac{1}{2} (r')^2 -\frac{\gamma}{r} + \frac{\ell^2}{2 r^2}.
$$
(This one is proportional to the asteroid's energy, kinetic and potential.)  We can rearrange this equation to yield
$$
r' = \pm \sqrt{e + \frac{\gamma}{r} - \frac{\ell^2}{2r^2}}
$$
which is a simple, separable first-order differential equation.  
You can see that by using the invariants, we have gone from a nasty set of coupled equations to an equation you probably could have figured out after your first week of class.
