Question

Use separation of variables to find, if possible, product solutions for the given partial differential equation. (Use the separation constant −λ ≠ 0. If not possible, enter IMPOSSIBLE.)
k∂^2u/∂x^2 - u = ∂u/∂t, k>0

Answer 1

Answer:

Following are the solution to this question:

Step-by-step explanation:

differentail equation:

$yu_{xy} + u = 0..........(1)$  

when $u(x,y) = X(x)Y(y)$

$u_x = \frac{\partial u}{\partial x} X'Y' \\\\ u_{xy} = \frac{\partial^2 u }{ \partial x\ \partial y}= X'Y'$

by substituting the value we get:

$\to \int \frac{Y'}{Y} = -\lambda \int \frac{1}{y} dy$

by solving the value:

$\to Y=c_3 y^{-\lambda}\\\\\therefore\ \ c_4=c_2C_3\\\\\to u(x,y) = c_4 e^{\frac{x}{\lambda}} y^{-\lambda}$