I try to solve the pde $u_x = c_0u + c_1(x,y)$, where $c_0$ is a constant, under the initial condition $u(0,y) = y$.
I use integrating factor on x to solve it. I get after some steps:
$$d(e^{-c_0x}u) = e^{-c_0x}c_1(x,y)dx$$
integrating;
$$e^{-c_0x}u + C = \int e^{-c_0x}c_1(x,y)dx$$
I'm stuck here beacuse I do not know how I can evaluate initial condition. Further, which variables appear in C?…
Best Answer
You have to take into account the integration interval/boundaries when going from the first line to the second one, such that : $$ \left.\color{white}{\frac{1}{1}}e^{-c_0x}u\,\right|_{u=u(0,y)}^{u=u(x,y)} = \int_0^x e^{-c_0x'}c_1(x',y) \,\mathrm{d}x' $$ hence $$ u(x,y) = e^{c_0x} \left(y + \int_0^x e^{-c_0x'}c_1(x',y) \,\mathrm{d}x'\right) $$ We see that the boundary condition $u(0,y) = y$ is indeed satisfied.