# Combining two partial differential equations into one

I have the equation $$\frac {\partial v}{\partial t}= \gamma \left(1-\frac{v}{v_0}\right)+\alpha \left(1-\frac{v}{v_0}\right)\rho-\beta (\rho-\rho_0)$$ and the mass conservation equation $$\frac{\partial\rho}{\partial t}+\frac{\partial(v\rho)}{\partial x}=0$$ where $v \equiv v(x,t)$ and $\rho \equiv \rho(x,t)$.

I'd like to combine these two to obtain an equation for $\rho$, eliminating $v$, but I can not see how.

## 1 Answer

The most straightforward, but tedious, way is to generate six equations by computing derivatives of the original equations wrt $x$ and $t$. You can then express them in matrix form $$\begin{bmatrix} A_{11} & A_{12} & A_{13} & A_{14} & A_{15} & A_{16} \\ A_{21} & A_{22} & A_{23} & A_{24} & A_{25} & A_{26} \\ \dots \\ A_{61} & A_{62} & A_{63} & A_{64} & A_{65} & A_{66} \end{bmatrix} \begin{bmatrix} v \\ \frac{\partial v}{\partial x} \\ \frac{\partial v}{\partial t}\\ \frac{\partial^2 v}{\partial x^2} \\ \frac{\partial^2 v}{\partial x\partial t} \\\frac{\partial^2 v}{\partial t^2} \end{bmatrix} = \begin{bmatrix} C_1 \\ C_2 \\ C_3 \\ C_4\\ C_5 \\ C_6 \end{bmatrix}$$ Solving for the unknowns will give you expressions that contain only $\rho$ and its derivatives that can be used to eliminate $v$ from your equations.

• well... that worked, but I got an equation 12 lines long. – Matias Morant Apr 11 '14 at 3:06
• Now all you have to do is simplify the equation :) – Biswajit Banerjee Apr 11 '14 at 3:12
• hahaha are you kidding?? that impossible Biswajit Banerjee! thanks I had not seen this method before anyway – Matias Morant Apr 11 '14 at 3:23
• You can simplify things slightly if you assume that $\rho \equiv \rho(x)$, then all the $\partial\rho/\partial t$ terms become zero. – Biswajit Banerjee Apr 11 '14 at 3:33
• Alternatively, you can try to repeat the process after multiplying the first equation by $\rho$ on both sides and see if terms drop out. – Biswajit Banerjee Apr 11 '14 at 3:43