math(galerkin,collocation)

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can some one plz help
how to make trial function for solving equations in collocation and galerkin methods which satisfy the given boundary conditions.
example y"+y=3*square(x) with y(0)=0 and y(2)=3.5.
is there some method of making trial func or we have to make it through intuition..

secondly how to solve equations with neumann conditions y"-(x+1)y=-e^-x(x^2-x+2) subject to neumann cond y'(2)=0,y'(4)=-0.036631

also if someone can giv me the link to and resource where i can get solved examples for my better understanding of these topics.
best regards n hoping to listen soon!

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