Question

Write a DE in the form dy/dt=ay+b ;y=2/3 t->0

1) dy/dt=a(y+b/a)
2)dy/(y+b/a)=a dt
I understand why it integrates as but do not understand why the there is a -ln(c) and not +c on both sids.
3)ln(y+b/a)-ln(c)=a*t

Answer 1

Can you take a snapshot of the original problem?

Answer 2

minus ln(c) is still subtract a constant
why can't it take this form?

Answer 3

first , if you have +c1 and +c2 on both sides, you can combine them into , for example, +c2-c1
and rename that as C (an arbitrary constant)

so people usually put the +C on only one side

Answer 4

Why is it ln(c). It does not seem very intuitive.

Answer 5

I think they wrote the constant as ln(c)
so they can write the left hand side as one term

Answer 6

\[\ln(y+\frac{b}{a})-\ln(c) = \ln(\frac{y+\frac{b}{a}}{c})\]

Answer 7

second, ln(C) is also arbitrary.
they are doing that because usually what we do is
ln y = x + C
make each side the exponent of e
y = e^(x+C)
y= e^C * e^x
and e^C is renamed an arbitrary constant , A (for example)
y = A e^x

Answer 8

Oh ok!!!

Answer 9

the idea is if you use ln(C) rather than c
you can do some algebra, and simplify the answer

Answer 10

I thought there was some hidden rule

Answer 11

That is just something that you have to train yourself to see?!

Answer 12

is ln(c) more of a substitute for C so that you can simplify easier?

Answer 13

yes the C took on the form ln(c) just to write things prettier like

Answer 14

THANK YOU EVERYONE!!!