3. Consider the linear partial differential equation \frac{\partial^{2} \phi}{\partial x^{2}}=\frac{\partial^{2} \phi}{\partial y^{2}}+2 \frac{\partial \phi}{\partial y} for a function (x, y) in the squ are 0 looking for a separable solution of the form o = X(x)Y(y) show that X and Ysatisfy respectively, for some separation constant k. (h) If we have the boun dary conditions \phi(0, y)=\phi(1, y)=\phi(x, 0)=0, \quad(* *) s how that X(x)=A_{n} \sin (n \pi x), \quad n=1,2,3, \ldots for A, constant, and that the most general solution to (*) subject to these boundary conditions is given by \phi(x, y)=\sum_{n=1}^{\infty} C_{n} \sin (n \pi x) e^{-y} \sin \left(\sqrt{n^{2} \pi^{2}-1} y\right) (c) Find the solution to the PDE (*) with boundary conditions (**) that additional ly satisfies (x, 1) = sin(Tr) and show that in this case the square.= 0 for only one value of y inside X^{\prime \prime}-k X=0 \quad \text { and } \quad Y^{\prime \prime}+2 Y^{r}-k Y=0
looking for a separable solution of the form o = X(x)Y(y) show that X and Ysatisfy respectively, for some separation constant k. (h) If we have the boun dary conditions \phi(0, y)=\phi(1, y)=\phi(x, 0)=0, \quad(* *) s how that X(x)=A_{n} \sin (n \pi x), \quad n=1,2,3, \ldots for A, constant, and that the most general solution to (*) subject to these boundary conditions is given by \phi(x, y)=\sum_{n=1}^{\infty} C_{n} \sin (n \pi x) e^{-y} \sin \left(\sqrt{n^{2} \pi^{2}-1} y\right) (c) Find the solution to the PDE (*) with boundary conditions (**) that additional ly satisfies (x, 1) = sin(Tr) and show that in this case the square.= 0 for only one value of y inside X^{\prime \prime}-k X=0 \quad \text { and } \quad Y^{\prime \prime}+2 Y^{r}-k Y=0
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