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leopard
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Homework Statement
The function u(x,t) satisfies the equation
(1) [tex]u_{xx}[/tex] = [tex]u_{tt}[/tex] for 0 < x < pi, t > 0
and the boundary conditions
(2) [tex]u_x[/tex](0,t) = [tex]u_x[/tex](pi, t) = 0
Show that (1) and (2) satisfy the superposition principle.
2. The attempt at a solution
I let w(x,t) = au(x,t) + bv(x,t) for two constants a and b.
[tex]w_{tt}[/tex] = [tex]au_{tt}[/tex] + [tex]bv_{tt}[/tex] = [tex]au_{xx}[/tex] + [tex]bv_{xx}[/tex] = [tex]cw_{xx}[/tex], where c is a constant
Have I now showed that w(x,t) satisfies (1)? [tex]w_{xx}[/tex] is not equal to [tex]w_{tt}[/tex] unless c is 1...
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