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Seperation of Variables/Integrating Factor Method
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[QUOTE="_N3WTON_, post: 4852510, member: 507278"] [h2]Homework Statement [/h2] The first order linear equation of the form: [itex]\frac{dy}{dx} + ay = b [/itex] where a and b are constants, can be solved both by the integrating factor method and by separation of variables. Solve the equation using both methods to see that you get the same solution. [h2]Homework Equations[/h2] Separation of variables and Integrating factor equations [h2]The Attempt at a Solution[/h2] I am 80% sure that I have correctly found the solution using both methods; however, I am having trouble equating the two solutions. This is what I have done so far, first I will show using the integrating factor method, followed by separation of variables: [itex] p(x) = a [/itex] [itex] u(x) = e^{ax} [/itex] [itex] \frac{d}{dx}(e^{ax} y(x)) = be^{ax} [/itex] [itex] \int\frac{d}{dx}(e^{ax} y(x))\,dx = \int be^{ax}\, dx [/itex] [itex] e^{ax} y(x) = abe^{ax} + C [/itex] [itex] y(x) = ab + \frac{C}{e^ax}[/itex] Now again using the separation of variables method: [itex] \frac{dy}{dx} = b-ay [/itex] [itex] \frac{dy}{y} = (b-a)dx [/itex] [itex] \int\frac{dy}{y} = \int b-a\, dx [/itex] [itex] ln(y) = bx - ax + C [/itex] [itex] y = e^{bx} - e^{ax} + e^C [/itex] [/QUOTE]
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Seperation of Variables/Integrating Factor Method
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