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Numerade Educator

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Problem 9 Medium Difficulty

Solve the differential equation.
$ \frac {dp}{dt} = t^2 - p + t^2 - 1 $

Answer

$$p=K e^{t^{3} / 3-t}-1$$

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Video Transcript

this question asks us to solve the differential equation. DP over. GT is T squared P minus P plus T squared minus one. Okay, first up, let's put stuff into factored forms than we can cancel off terms. Soapy times T squared, minus wanna is our first part. And then we obviously can't factor t squared minus one with any more variables. Now we want all the peace stuff in the left hand side and all the t's stuff on the right hand side. So do some manipulation. So you have dp over people. Swan, the left hand side on that T squared minus one D two. On the right hand side, it is very important that we put each variable on separate sides so we cannot integrate on the left hand side, we have natural log of people swan. And then we have minus natural FCC being our constant of integration is equivalent to we had t squared them, increase the exploited by one divide by the new exponents on the negative one integrates to negative t This simplifies to people's one oversee is e to the T cubed minus three t divide by three. And then lastly we want to see on the right hand side. So we have people. Swan. There is multiplying both sides by the sea in order to get our final simplifying solution.