# Math Help - simple ODE

1. ## simple ODE

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?

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

2. 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 $y$

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

3. I get

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