Agent_A
  • Agent_A
Differential Equations problem. Solve the initial Value Problem: (see the given, below) I have the solution, and I know that we have to use integration by parts, but I'd like a better (clearer) solution, please. Just send me the whole thing. I want to see the way you solve it, in one run. Thanks!
Mathematics
schrodinger
  • schrodinger
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Agent_A
  • Agent_A
\[\frac{ dy }{ dt } = y + 2t\] \[y(0) = -2\]
ganeshie8
  • ganeshie8
As a start, multiply \(e^{-t}\) through out
ganeshie8
  • ganeshie8
\(\dfrac{ dy }{ dt } = y + 2t\) \(\dfrac{ dy }{ dt } -y = 2t\) \(\color{red}{e^{-t}}\dfrac{ dy }{ dt } - \color{red}{e^{-t}}y = \color{red}{e^{-t}}2t\) Now, do you notice anythign special about the left hand side ?

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Jhannybean
  • Jhannybean
Can this form be defined as \(\dfrac{dy}{dt} - P\cdot y = Q\) ?
Jhannybean
  • Jhannybean
Oh I kind of see it now.
ganeshie8
  • ganeshie8
Yes, it's indeed a linear equation..
Astrophysics
  • Astrophysics
Note the integrating factor is \[e^{-t}\] now what happens when you take the derivative of this \[(e^{-t}y)'\]

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