simple ODE

**Hoover glasses** · September 16th 2008, 03:55 PM

I'm taking a first course in DE. Is this problem in the form $\frac{dy}{dt}+ay=g(t)$ ? How do I solve it?

$y\prime + 3y = t + e^{-2t}$

**Jhevon** · September 16th 2008, 05:52 PM

Originally Posted by Hoover glasses

I'm taking a first course in DE. Is this problem in the form $\frac{dy}{dt}+ay=g(t)$ ? How do I solve it?

$y\prime + 3y = t + e^{-2t}$

$y' + ay = g(t)$ .........................multiply through by $e^{at}$

$\Rightarrow e^{at}y' + ae^{at}y = e^{at}g(t)$ .........simplify the left hand side

$\Rightarrow (e^{at}y)' = e^{at}g(t)$ ..................integrate both sides

$\Rightarrow e^{at}y = \int e^{at}g(t)~dt$ .............solve for $y$

$\Rightarrow y = \frac {\int e^{at}g(t)~dt}{e^{at}}$

**Hoover glasses** · September 16th 2008, 11:52 PM

I get

$y = \frac{t}{3} -1/9 +e^{-2t} + ce^{-3t}$

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