Find all complex exponential solutions u(t, x)=e^ωt+k x of the beam equation (∂^2 u)/(∂t^2)=(∂^4

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Step 1:u000AStart by substituting the proposed solution u005C( u(t, x) u003D e^{u005Comega t + kx} u005C) into the b

Find all complex exponential solutions $u(t, x)=e^{\omega t+k x}$ of the beam equation $\frac{\partial^2 u}{\partial t^2}=\frac{\partial^4 u}{\partial x^4}$. How many different real solutions can you produce?