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Numerade Educator

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Problem 13 Medium Difficulty

Solve the differential equation.
$ t^2 \frac {dy}{dt} + 3ty = \sqrt {1 + t^2}, t > 0 $

Answer

$$y=\frac{1}{3} t^{-3}\left(1+t^{2}\right)^{3 / 2}+C t^{-3}$$

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Video Transcript

In order to solve this equation, we must first get into the standard form. Why? Prime plus p of x times. Why is Q backs and orgy this? We must divide by t squared to get this into the standard form that I just mentioned. So as I said, I'm dividing by t squared. Okay, Now that we've got this, we know we must determine the integrating factor. Each of the integral of three over tee times D t is the same thing has eaten the three, not a log of tea which is the same thing as eat the natural of t cubed. Now we know eats the natural log is once this the same thing as t cubed. We have just determined are integrating factor. Okay, Now we must multiply both sides. In other words, all terms by the integrating factor that we just determined in order to know integrate both sides and sulfur Why we have t cubed. Why is the integral of tea times one plus she squared d t which gives us t cubed. Why is one third times one plus t squared 23 over to pussy We use the power room or to do this other words. We increased our expert by one and we then divided by the new exponents. Lastly, we only want this in terms of why not any sort of co fishing in front? That isn't one, which means we must divide by t cubed for the terms in order to derive our final solution plus c times t to the negative three which the same thing is pussy over too cute.