Solve the equation using green's function $-u''=f(x), 0<x<1, u(0)=u(1)=0$
I got the solution as being $u(x)=\frac{x^2}{2}(x-1)f(x)+xf(x)(\frac{1}{2}-\frac{x^2}{2}-1+x$.
Which is wrong. Can anyone see where i went wrong?
2026-03-28 15:26:05.1774711565
Solve the equation: $-u''=f(x), 0<x<1, u(0)=u(1)=0$
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The next to last formula is wrong, it should have $f(s)$ instead of $f(x)$, that is, $$ u(x)=\int_0^1G(x,s)f(s)\,ds=-\int_0^x (1-x)sf(s)\,ds-\int_x^1 x(1-s)f(s)\,ds $$ Without knowing $f$ no further evaluation of the integrals is possible.