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A technique to solve first order differential equations of the form

{{dy \over dx} + p(x)y = q(x)}.

  • Calculate the integration factor {\rho (x) = e^{\int p(x) dx}}.
  • Multiply each side of the equation by {\rho (x)}
  • Integrate both sides of the equation to get {\rho(x)y(x) = \int \rho(x)q(x) dx}.
  • Sove for y(x).


Example

{{dy \over dx} - y = {11 \over 8}e^{-x/3}}

{\Rightarrow \rho(x) = e^{\int (-1) dx} = e^{-x}}

{\Rightarrow e^{-x}{dy \over dx} - e^{-x}y = {11 \over 8}e^{-4x/3}}

{\Rightarrow e^{-x}y = \int {11 \over 8}e^{-4x/3} dx = {-33 \over 32} e^{(-4x/3)} + C}

{\Rightarrow y(x) = {-33 \over 32} e^{(-x/3)} + Ce^{x}}

External References

http://en.wikipedia.org/wiki/Integration_factor

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