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I'm trying to solve the following system of PDEs with the method of characteristics:

$\frac{\partial T}{\partial t}(t,x) = \gamma(t,x) - \psi(t,x) T(t,x) + \kappa(t,x) u(t,x) \\ \frac{\partial u}{\partial t}(t,x) + \beta (t,x) \frac{\partial u}{\partial x} (t,x) = \varrho (t,x)(T(t,x) - u(t,x))$

Making $p_{1} = \frac{\partial T}{\partial t}(t,x)$, $q_{1} = \frac{\partial T}{\partial x}(t,x) = 0$, $p_{2} = \frac{\partial u}{\partial t}(t,x)$, $q_{2} = \frac{\partial u}{\partial x} (t,x)$ , and applying the method of characteristics in both equations it is obtained:

$\dot{T}_{1} = p_{1} = - \gamma(t,x) - \psi(t,x) T(t,x) + \kappa(t,x) u(t,x)\\ \dot{x}_{1} = 0 \\ \dot{t}_{1} = 1 \\ \dot{p}_{1} = - p_{1}\psi(t,x)\\ \dot{q}_{1} = 0$


$\dot{u}_{2} = p_{2} + \beta(t,x)q_{2} = \varrho(t,x) (T(t,x)-u(t,x))\\ \dot{x}_{2} = \beta(t,x)\\ \dot(t)_{2} = 1 \\ \dot{p}_{2} = p_{2} \varrho(t,x)\\ \dot{q}_{2} = -q_{2} \varrho(t,x)$

But now I do not know what to do with these ODEs. Anyone knows if I'm right and how can I solve this set of ODEs?

best regards,

Gustavo

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