Solve PDE: Find F to Satisfy \lambda F + (\frac{\partial F}{\partial y})^{2}=0

[Karlisbad]

Solving this PDE :(

Hello i have a question about this..let be a function $$F(x(t),y(t),z(t),t)$$ then if we use the "total derivative" respect to t and partial derivatives..could we find an F so it satisfies:

$$\frac{d (\frac{\partial F}{\partial x})}{dt}+\lambda F + (\frac{\partial F}{\partial y})^{2}=0$$ ??

how could you solve that ??.. my big problem is that this involves "total" and partial derivatives respect to x and y all mixed up.

 

[Amr Morsi]

Karlis,

Since the equation includes total derivative with respect to t, then x, y and z are functions of t as already included in the argument of F; X(t) ....

However, you still can solve it, but in terms of dx/dt, dy/dt, dz/dt ... etc. But, here, only function in dx/dt, dy/dt and dz/dt.

Try this:
d[T(r>,t)]/dt=dx/dt*p[T(r>,t)]/px+dy/dt*p[T(r>,t)]/py+dz/dt*p[T(r>,t)]/pz.
where p/px is the partial derivative.
This can be done by the rule of differentials.
Here, T(r>,t)=pF/px

You will get a normal partial differential function in x, y and z with 3 time-dependent functions (considered to be constants in the equation). Solve it, if this form has an analytical solution (or any other sort) in PDEs.


Engineer\ Amr Morsi.