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What are the integrating factors for the following differential equations?$$\frac{d y}{d t}+3 t y=e^{t} y$$
$\mu(x)=e^{\left(\frac{3 t^{2}}{2}-e^{t}\right)}$
Calculus 2 / BC
Chapter 4
Introduction to Differential Equations
Section 5
First-order Linear Equations
Differential Equations
Missouri State University
Baylor University
Idaho State University
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high in this video, we are going to fi and integrating factor fall this equation. Why prime plus treed t y equals e to the T. Why? First, we arrange it into a standard form wish simply just more e to the t y to the left side and pulling. Why? I'll get this equation so you can see that p of x ys sorry. Here it would be p off t. It is pre t minus eternity and our so our integrating factor is power Bi. Yeah, great p and a B two wish can be done easily. So let's do this integrate thing three d e to the two. I do too. Now we can breathe. We can break this into two parts because it joined by addition and minus So the first part is gonna go treaty ditty the second parties. You go e to the t day t here, so we get t square times tree over too one over to appear from the integration and yeah, you probably know from the material that in big Red e to the t t t will give the same function so e to the t and this is our This is what I need to go. Petey! Petey! Give. Thus we have integrating fact the ask E to day tree over to T Square minus e to the T. Yeah, um, don't confuse this with more off T. Uh, this is E to the t and it stay like that. If you convinced me not off t you may like shrinking doll into into tea or something which is not correct. So let let it stay there. And here we have it. This is integrating factor. Thank you.
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