Ordinary equations help

• September 24th 2008, 10:59 PM
kirbyiwaki
Ordinary equations help
It happens that I'm not good with integrals, and our teacher is a good teacher, but has a little problem: he uses easy examples but gives us difficult homework... I've managed to do most of it, but have trouble with last ones... can anyone help me please? I'm desperate (Crying)!!!

I-Solve the next lineal equations:

4.-
http://img228.imageshack.us/img228/7734/001on8.gif
II- Find the general solution for the next exact ordinary equations:

http://img228.imageshack.us/img228/9266/002wn7.gif

(I know the first one of these is not exact... but can it be solved with an integrating factor?... and the last one... I know it says general solution too... do you think is a trap from my teacher [last time he said "one of the problems didn't have any solution", so I get that wrong...])

Also... procedures will be highly appreciated n__n"

And, after the answers... can someone teach me the art of EDO?... I'll be very grateful, thanks in advance.
• September 25th 2008, 12:49 AM
Peritus
$
\begin{gathered}
\frac{{dP}}
{{dt}} + 2tP = P + 4t - 2 \hfill \\
\Leftrightarrow \frac{{dP}}
{{dt}} + (2t - 1)P = 4t - 2 \hfill \\
\Leftrightarrow e^{t^2 - t} \frac{{dP}}
{{dt}} + e^{t^2 - t} (2t - 1)P = e^{t^2 - t} \left( {4t - 2} \right) \hfill \\
\end{gathered}
$

$
\begin{gathered}
\Leftrightarrow \frac{d}
{{dt}}\left( {e^{t^2 - t} P} \right) = e^{t^2 - t} \left( {4t - 2} \right) \hfill \\
\Leftrightarrow e^{t^2 - t} P = \int {e^{t^2 - t} \left( {4t - 2} \right)dt = 2} \int {\left( {2t - 1} \right)e^{t^2 - t} dt = 2} e^{t^2 - t} + C \hfill \\
\Leftrightarrow P(t) = 2 + Ce^{t - t^2 } \hfill \\
\end{gathered}
$

Exact First-Order Ordinary Differential Equation -- from Wolfram MathWorld
• September 25th 2008, 01:38 AM
kirbyiwaki
Thank you~!!! I had it almost done (via general formula, tough), but the integration ruined me u.u... thanks again ^^!