## Agent_A one year ago 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!

1. Agent_A

$\frac{ dy }{ dt } = y + 2t$ $y(0) = -2$

2. ganeshie8

As a start, multiply $$e^{-t}$$ through out

3. 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 ?

4. anonymous

Can this form be defined as $$\dfrac{dy}{dt} - P\cdot y = Q$$ ?

5. anonymous

Oh I kind of see it now.

6. ganeshie8

Yes, it's indeed a linear equation..

7. Astrophysics

Note the integrating factor is $e^{-t}$ now what happens when you take the derivative of this $(e^{-t}y)'$