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I am trying to learn about solving boundary value problems, but I stuck when I came to finding BVP of $1$D Poisson equation on $[0,1]$: $$ \begin{cases} \dfrac{\mathrm{d}^2}{\mathrm{d}x^2}u(x)=-g(x) \\ u(0)= a \\ u(1)= b \end{cases} $$ where $g(x)=−6\pi \cos(3\pi x)+9\pi^2x \sin(3\pi x)$ and $a=b=0$.

Should I get the integral of given $g$ two times and then plug in the $u(0)$ and $u(1)$ values to find out the values of the two integration constants $c_1$ and $c_2$? Or am I supposed to do something else? Also, how can I start solving it numerically by discretization for any $h$ value? Any hint or tip is appreciated.

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