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## Homework Statement

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 seperation of variables. Solve the equation using both methods to see that you get the same solution.

## Homework Equations

Separation of variables and Integrating factor equations

## The Attempt at a Solution

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]