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Solve y''(x)=f(x),y(0)=A,y'(0)=B\,

Integrate both sides with respect x.

y'(x)-y'(0)=\int _{0}^{x}f(s)\,ds\,

y'(x)-B=\int _{0}^{x}f(s)\,ds\,

y(x)-y(0)=B\int _{0}^{x}\,du+\int _{0}^{x}\int _{0}^{u}f(s)\,ds\,du\,

y(x)=y(0)+Bx+\int _{0}^{x}\int _{0}^{u}f(s)\,ds\,du\,

y(x)=A+Bx+\int _{0}^{x}\int _{0}^{u}f(s)\,ds\,du\,

y(x)=A+Bx+\int _{0}^{x}f(s)(x-s)\,ds\,

Integral Equations

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