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I am just starting to study PDE and don't know how solve the Dirichlet problem on the line. I read PDE book, but they discuss in more than one dimension by using separation of variable. How does one do it in one dimension. I know you just get $\sin(nx)$, but how do you get that?

In fact, I am even more confused about the separation of variables technique on the plane.

d$u_{xx}+u_{yy}=0 $ Using separation of variables, one gets $u(x,y)=X(x)Y(y)$, which leads to the conclusion that $\frac{d^{2}X}{dx^{2}}+\lambda X=0$

and $ \frac{d^{2}Y}{dy^{2}}-\lambda Y=0 $.

Should it just be $\frac{d^{2}X}{dx^{2}}=0$ and analogously for $Y$?

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