`t(dx)/(dt)-x=3`

We want all x's on one side, all the terms in t on the other.

`t(dx)/(dt)=x+3`

Divide both sides by `(x + 3)` and multiply both sides by `dt`:

`t(dx)/(x+3)=dt`

Next, divide both sides by t.

`(dx)/(x+3)=(dt)/t`

Integrate both sides.

`int(dx)/(x+3)=int(dt)/t`

Important: dx and dt must be on top of the fraction (i.e. in the numerator).

`ln|x+3|=ln|t|+C`

Take "e to power of both sides":

`x+3=e^(ln|t|+C)=e^(ln|t|)e^C=Kt`

(We let `e^C=K`, constant)

So `x = Kt - 3`.

Typical solution and graph

We let `K = 7`, for illustration):

1234551015202530-5txOpen image in a new page

Typical solution graph `x=7t-3`.

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