Determine whether each of the following functions is a solution of Laplace's equation uxx + uyy = 0. (Select all that apply.)
u = e^(-x) cosy - e^(-y) cosx
u = ln(x^2+y^2)
u = x^3 + 3xy^2
u = x^2 - y^2
u = sinxcoshy + cosxsinh
u = x^2 + y^2
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Determine whether each of the following functions is a solution of Laplace's equation uxx + uyy = 0. (Select all that apply.)
u = e^(-x) cosy - e^(-y) cosx
u = ln(x^2+y^2)
u = x^3 + 3xy^2
u = x^2 - y^2
u = sinxcoshy + cosxsinh
u = x^2 + y^2
![Consider the equation y'' - y' - 2y = 0. (a) Show that y1(t) = e^-t and y2(t) = e^2t form a fundamental set of solutions. (b) Let y3(t) = -2e^2t , y4(t) = y1(t) + 2y2(t), and y5(t) = 2y1(t) - 2y3(t). Are y3(t), y4(t), and y5(t) also solutions of the given differential equation? (c) Determine whether each of the following pairs forms a fundamental set of solutions: [y1(t), y3(t)]; [y2(t), y3(t)]; [y1(t), y4(t)]; [y4(t), y5(t)].](https://www.doubtrix.com/js/ckeditor/filemanager/connectors/php/editor/1685797784-Q22.png)
