# Solve the differential equation

Discussion in 'Homework Help' started by poopscoop, Feb 5, 2014.

1. ### poopscoop Thread Starter Member

Dec 12, 2012
139
16
(t$^{2}$-24t+108)dy/dt=y

Separate variables:

1/((t-18)(t-6))=1/y

Integrate, partial fraction decomposition (Checked with Wolfram alpha, these are correct):

1/(12(t-18))-1/(12(t-6))=1/y

ln((t-18)/(t-6))+c=12ln(y)

-----Now unsure
Raise everything to e and move the 12 over (Do I need to treat that 12 as an exponent after I raise everything to e?):

(t-18/(12(t-6))+c=y

Initial condition y(12)=1
thus c= 13/12

The are two other parts to this problem, asking what the range of t values the solution is valid for, which according to the variable-separated differential is 6 through 18. On there I am correct. The second part asks what the solutions approach as t approaches these limits, (infinity and 0) where I'm also correct.

So something I'm doing is wrong, but not fundamentally wrong enough to screw up the whole question.

2. ### poopscoop Thread Starter Member

Dec 12, 2012
139
16
Oh wow, latex did not work for me. Editing.

3. ### WBahn Moderator

Mar 31, 2012
20,247
5,758
What happen to the constant c when you exponentiate everything?

e.g.

y = ln(x) + c

what is e^y?

To see what happens to the 12, consider a simpler case in which you have

y = 12x

what is e^y?

Now change that to

y = 12ln(x)

what is e^y