Hello, I am trying to solve the following equation:
\frac{\partial x}{\partial t} = A \frac{ \partial x}{\partial y} + B \frac{\partial^2 x}{\partial y^2}
I know how to solve the diffusion equation (i.e. no dx/dy term), but that method doesn't work here. I tried to go with the LaPlace...
I haven't done this in ages, and I'm having trouble recalling how to factor a higher order polynomial. I almost always do this graphically, but for this case I'm interested in an algebraic solution. Specifically, I'm looking at ax + x^3 - x^5 = 0 (with a = an integer >0.)
Clearly 0 is one...
In my continued pursuit of isothermal equation of state solutions, I've come upon a very simple integral I can't recall how to do, and was looking for assistance.
It is:
du = \int \frac{a}{T^{0.5}v(v+b)}dv
where a, T and b are constants. so:
du = \frac{a}{T^{0.5}} \int \frac{1}{v(v+b)}dv...
I'm trying to integrate the Van der Waals equation of state for an isothermal problem, but based on my results I think I'm doing a bit of simple calculus wrong and hope someone here can help.
P = \int \frac{RT}{v-b} dv
where R, b and T are known constants.
I tried to do a u-substitution...
Simple Diff Eq Help
I am trying to solve the following Diff Eq:
\frac{d^2x}{dy^2}+(\frac{y}{2}-\frac{1}{y})\frac{dx}{dy}=0
I tried to solve by setting \frac{dx}{dy}=z
so: \frac{dz}{dy}+(\frac{y}{2}-\frac{1}{y})z=0
I know the general solution to this is...
In my struggles to understand index notation, I am trying to figure out how my book came up with the following transformation.
\frac {D \omega}{Dt} \cdot \omega = \omega_j \partial_j v_i \cdot \omega + \nu \partial_j \partial_j \omega_i \cdot \omega
=
\frac {D \frac{\omega^2}{2}}{Dt} =...
I am playing around with learning index notation for tensors, and I came across the following where C is a 0th order tensor:
E_{ijk} \partial_j \partial_k C = 0
I believe this equates to \nabla \times \nabla C. I don't understand why this comes out to 0. Any ideas?
Also, I am trying...
I am familiar with how to solve a second order, non-homogenous DE with constants, i.e.
\frac {\partial^2X(t)}{\partial t^2} + \frac{\partial X(t)}{\partial t} = C
by first solving the homogenous eqn, then setting the equation equal to a constant, yielding a sol'n of
X(t)= Ae^{0}+...
I've reduced a portion of the Navier Stokes to solve a flow problem, and am left with the following ODE:
u (\frac {\partial^2 Vz} {\partial^2 r}) + \frac {r} {u}\frac {\partial Vz} {\partial r} = 0
I tried to solve this equation by assuming a power law solution with
Vz = Cr^n
Which...