# Tagged Questions

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### How to solve a system of first-order partial differential equations?

I have a system of first-order partial differential equations. \begin{align} & -\frac{a}{a_1} \frac{\partial P_{12}}{\partial a} - \frac{a b}{a_1 b_1} \frac{\partial P_{12}}{\partial b} + ...
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### Solve a system of equations.

I have a system of equations: \begin{align} & x_{21} (\frac{\partial}{\partial x_{11}}f_{1111})( x_{11} , x_{21}, y_{11} , y_{21} ) + \frac{y_{21}}{x_{11}^2} (\frac{\partial}{\partial ...
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### Pair of PDEs to be solved together

I have the following pair of equations to be solved together to find the functions $H_{x}$ and $H_{y}$, which are the components of a vector $\bar{H}(x,y)=H_{x}(x,y)\hat{x}+H_{y}(x,y)\hat{y}$ in ...
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### System of ODEs and DAE system

Let us consider the following system of ODEs: $$y' = f(y,z),\quad z' = g(y,z),\quad y(0) = y_0,\;z(0)=z_0$$ and the following one: $$y' = f(y,z),\quad 0 = g(y,z), \quad y(0) = y_0.$$ $f$ and $g$ ...
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### Methods of characteristic for system of first order linear hyperbolic partial differential equations: reference and examples

I would like to understand a few points on the methods of characteristics used to resolve a system of coupled, linear first order partial differential equation (of the hyperbolic type). Some example ...
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### For a system of PDEs, how many equations are needed generally for the system to have unique solution?

For an algebraic system of equations or a system of ordinary differential equations the following rule holds:(right?) the total number of unknown variables must be equal to the number of equations ...
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### System of linear differential equations eigenproblem

In the Smith and Young 2001 paper on the Barotropic tide they have the governing equations... \begin{array}{rcl} u_t-f_0v+p_x & = & 0 \\v_t+f_0u+p_y & = & 0 \\p_z &= &b ...
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### On a system of PDE

I would like to know what is the set of solutions to the following PDE. I think it consists of just constants, but I need help to prove. Let $f_1(p_1,p_2)$ and $f_2(p_1,p_2)$ be two functions. The ...
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### Constructing a Distributional Solution to the Inhomogeneous C.R. Equations

The question is to find a fundamental solution to the system of equations in $\mathbb{R}^{2}$ \begin{array}{l} u_{x}-v_{y}=f\\ u_{y}+v_{x}=g\end{array} and to express the answer as a $2\times2$ ...
It is fairly straight forward to solve linear 1st order PDEs by the method of characteristics. For example, if $\partial_tf+a\partial_xf=bf$ , we have that $\dfrac{df}{dt}=bf$ on the characteristic ...