IEVS2

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Solve u(x)=x+\int _{0}^{x}(x-y)u(y)\,dy

In this problem f(x)=x,\,\,\,k(x-y)=x-y\,

Take the Laplace transform of both sides.

{\mathcal  {L}}u={\mathcal  {L}}x+{\mathcal  {L}}x{\mathcal  {L}}u\,

{\mathcal  {L}}u={\frac  {{\mathcal  {L}}x}{1-{\mathcal  {L}}x}}\,, where {\mathcal  {L}}x={\frac  {1}{s^{2}}}\,

The solution is

u(x)={\mathcal  {L}}^{{-1}}\left({\frac  {1}{s^{2}-1}}\right)={\frac  {1}{2}}\left(e^{x}-e^{{-x}}\right)\,

Integral Equations

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