FANDOM


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