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Solve the differential equation.

$ t \ln t \frac {dr}{dt} + r = te^t $

$$

r=\frac{e^{t}+C}{\ln t}

$$

Differential Equations

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Campbell University

University of Nottingham

Idaho State University

in order to solve this equation, me West first put it into the standard formula. Why prime plus y p of X is Q of X. In other words, we must divide both sides by t times the natural log of tea. So we have one divided by t, not to a log of tea. Times are plus deep are over. DT is equivalent to eat the tea over natural of teeth. Now second stuff is integrating factor each the integral off one over t natural order tea again, that's what we just divided each of these terms. But integrate this. We get eat the natural log of natural with T. Remember each the natural August 1, which means our intruding factor is simply a natural log of teeth. Now this must be multiplied by each of these in visual terms, you'll probably notice that some stuff will end up being crossed out, which is good, because this means that it's simpler when we end up doing the integration on the right hand side. Okay, now that we have this, we know that we have our natural log of tea on the left hand side. Then we have the integral of each the T detail on the right hand sacked. This is actually a pretty straightforward and simple and to grow because the integral of each of the tea is literally just you the t remember, we must outer integrating factor. See, now, lastly, we know that we want we know that we want, um are by itself in this context, which means we're going to be dividing each of these terms by natural log of tea. Okay, in order to get our by itself, which means we now have our is eat the tea plus see some just combining these under the same denominator to make it easier to read.