# Thread: linear equation with integrating factor

1. ## linear equation with integrating factor

Hello,
I have been trying this problem for about an hour now and I just can't seem to figure it out:

dA/dt + 2A/(50+t) = 3

I found that the integrating factor = (50+t)^2 and so d/dt (50+t)^2(A) = 3(50+t)^2

but after this my problem falls apart... I can't seem to figure out how to go from there... I tried integrating both sides and factoring but that was not the right answer... Any help would be greatly appreciated! Thank you!

2. Originally Posted by collegestudent321
Hello,
I have been trying this problem for about an hour now and I just can't seem to figure it out:

dA/dt + 2A/(50+t) = 3

I found that the integrating factor = (50+t)^2
Until now, everything's fine.

Originally Posted by collegestudent321
and so d/dt (50+t)^2(A) = 3(50+t)^2
I'm not really sure what you did here. Maybe Latex could help me to understand.
but after this my problem falls apart... I can't seem to figure out how to go from there... I tried integrating both sides and factoring but that was not the right answer... Any help would be greatly appreciated! Thank you!
From it, multiply both side by the IF. Then integrate with respect to t. You'll reach $\displaystyle (50+t)^2y=3\int (50+t)^2 dt$.

3. Originally Posted by collegestudent321
Hello,
I have been trying this problem for about an hour now and I just can't seem to figure it out:

dA/dt + 2A/(50+t) = 3

I found that the integrating factor = (50+t)^2 and so d/dt (50+t)^2(A) = 3(50+t)^2

but after this my problem falls apart... I can't seem to figure out how to go from there... I tried integrating both sides and factoring but that was not the right answer... Any help would be greatly appreciated! Thank you!
$\displaystyle \frac{dA}{dt} + \frac{2A}{50 + t} = 3$

$\displaystyle (50 + t)^2\frac{dA}{dt} + 2(50 + t)A = 3(50 + t)^2$

$\displaystyle \frac{d}{dt}[(50 + t)^2A] = 3(50 + t)^2$

$\displaystyle (50 + t)^2A = \int{3(50 + t)^2\,dt}$

$\displaystyle (50 + t)^2A = (50 + t)^3 + C$

$\displaystyle A = 50 + t + \frac{C}{(50 + t)^2}$.

4. wow... I would have never guessed that. Thank you so much!

5. Originally Posted by collegestudent321
wow... I would have never guessed that. Thank you so much!
I suggest you to read the second post of this thread: http://www.mathhelpforum.com/math-he...-tutorial.html.
If you would have never guessed how to proceed after getting the IF, it means you didn't really understand the method itself. Chris' explanation of the method in 3 lines explain it very well. Have a look.