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I need to solve the PDE below using direct integration; $$ u_{xt} = cos(2t), (x>0, t>0), u(x,0)=x, u(0,t)=0 $$ But when I change the integration order I realize that the resulting u(x,t) function also changes. How can I do it?

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Integrate first wrt $x$. $$u_t=x\cos(2t)+g(t)$$ Then integrate wrt $t$. $$u=\frac x2\sin(2t)+G(t)+f(x)$$ Now substitute the boundary conditions in which will allow you to determine G and f.

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