# Search results

1. ### How to Solve dx/dt = Adx/dy + Bdx^2/dy^2

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...
2. ### Finding zeros

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...
3. ### Help with a simple integral

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...
4. ### Simple Integral Help

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...
5. ### Integration Help

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...
6. ### Index transformation

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} =...
7. ### Questions about Index Notation

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...
8. ### 2nd Order De Solution

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}+...
9. ### Solving a Simple ODE from the Navier-Stokes

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...