anonymous
  • anonymous
Solve (differential equations): dy/dt = y*e^3t + e^-t
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
is it a matter of setting up a dy and y-terms on one side and the dt and t-terms on the other?
watchmath
  • watchmath
Do you know about the integrating factor?
anonymous
  • anonymous
vaguely, but i'm not too sure on it yet

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watchmath
  • watchmath
For differential equation of the type \(y'+p(t)y=q(t)\) the trick is to multiply the equation by \(e^{\int p(t)\, dt}\)
anonymous
  • anonymous
so in this case, \[e ^{\int\limits_{}^{} e ^{3t}dt}\] ?
watchmath
  • watchmath
yes :)
watchmath
  • watchmath
well to be precise since you move the \(e^{3t}y\) to the left, your integrating factor is \(e^{\int -e^{3t}\,dt}\)
watchmath
  • watchmath
So that integrating factor is \(e^{-e^{3t}/3}\). Multiply the equation by that we have \(e^{-e^{3t}/3}y'-e^{-e^{3t}/3}e^{3t}y=e^{-e^{3t}/3}e^{-t}\) \((e^{-e^{3t}/3}y)'=e^{-t-e^{3t}/3}\) then integrate both sides.
anonymous
  • anonymous
how would that be done with the y' factor?
watchmath
  • watchmath
Notice that if we apply the the product rule to compute the derivative of \(e^{-e^{3t}/3}y\) we will get the previous step.
anonymous
  • anonymous
how would all that simplify? i'm not sure if i'm mixing up rules or what, but i'm coming up with far-out answers that don't simplify.
watchmath
  • watchmath
hmm what is derivative of \(e^{-e^{3t}/3}y\) with respect to \(t\)?
anonymous
  • anonymous
\[e ^{-e ^{3t}}y\] ?

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