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I have a problem in the form

[tex]\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}+e^{x}u=0[/tex]

with conditions

[tex]u(x,0)=u_0(x)[/tex]

[tex]u(0,t)=\int_{0}^{\infty}f(x)u(x,t)dx[/tex]

Im confused, because in first order PDE i require only 1 condition. How to solve this for two conditions?