Let&nbsp;f∈Co∞(Rn). Propose formulas of the form&nbsp;u(x)=∫&nbsp;&nbsp;f^(δ)eixδp(δ)dδto solve:i)&nbsp;∑j=1n∂4u(x)∂xj4+u(x)=f(x)andii)&nbsp;∑k,l&nbsp;∂4u(x)∂xk2∂xl2-2·∑k=1n∂2u(x)∂xk2+u(x)=f(x)

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Let&amp;nbsp;f∈Co∞(Rn). Propose formulas of the form&amp;nbsp;u(x)=∫&amp;nbsp;&amp;nbsp;f^(δ)eixδp(δ)dδto solve:i)&amp;nbsp;∑j=1n∂4u(x)∂xj4+u(x)=f(x)andii)&amp;nbsp;∑k,l&amp;nbsp;∂4u(x)∂xk2∂xl2-2·∑k=1n∂2u(x)∂xk2+u(x)=f(x)

Yaretzi Odom · Accepted Answer

Step 1For a general linear differential operator L[u] with constant coefficients you know that to get a particular solution toL[u]=ceiδ⋅xyou make a trial approach with u=Aeiδ⋅xand you get L[ei∂ts⋅x]=p(δ)eiδ⋅xSo A=cp(δ). If the right side is a sum of exponential terms,L[u]=∑k=1nckeiδk⋅xthen the particular solution is composed of the particular solutions for each term,up(x)=∑k=1nckp(δk)eiδk⋅x.Now go to the limit in some fashion, for instance via step functions, to apply an analogous argument when the right side is a function with a Fourier transform,L[u](x)=f(x)=∫Df(δ)^eiδ⋅x,dδ.

Let f \in C_{o}^{\infty} (\mathbb{R}^n). Propose formulas of the form

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