Question

DIFFERENTIAL EQUATION
find y given the differential equation (x+1)dy/dx = y with condition y(4)=10

Answer 1

Hmm this appears to be separable.
So let's move some things around.

Answer 2

There is a process that allows us to write the dx differential on the other side. For our purposes, we can simply think of this as multiplication, even though that's not exactly accurate. :)\[\large (x+1)dy=ydx\]

Answer 3

Now we'll move the y term over with the dy, and the x term over to the side with the dx.\[\large \frac{dy}{y}=\frac{dx}{x+1}\]

Answer 4

And now we integrate! Yay!

Answer 5

\[\large \int\limits \frac{dy}{y}=\int\limits \frac{dx}{x+1}\]

Answer 6

\[\large \ln y = \ln(x+1)+c\]Confused about anything yet?
Understand why we only write a +C on one side?

Answer 7

I approached the question wrong before

Answer 8

+C is what i need to solve right?

Answer 9

the initial condition

Answer 10

Yes, the initial condition allows you to find your C :)

Answer 11

We don't want to find this C though. We want to manipulate the problem a little more BEFORE we solve for C.

Answer 12

If we exponentiate both sides (Write them with a base of e) we get,\[\large e^{\ln y} = e^{\ln(x+1)+c}\]

Answer 13

the exponential and logarithm are INVERSE operations of one another, so on the left, they'll cancel out nicely.\[\large y = e^{\ln(x+1)+c}\]

Answer 14

then its y=(x+1)+c right?

Answer 15

No. the right side you can't pull that little tricky I'm afraid, because of the +c attached to the log.

Answer 16

ok thx for pointing thst out

Answer 17

We have to do a little something involving exponents before we can do that little magic trick.

Answer 18

Recall this exponential law?\[\huge a^{b+c} \quad = \quad a^b\cdot a^c\]

Answer 19

im guessin we do e^(ln(x+1) + e^c
or am i doing it wrong

Answer 20

e^(ln(x+1) * e^c

Answer 21

Applying this law lets us rewrite the right side,\[\huge y = e^{\ln(x+1)}\cdot e^c\]You're on the right track, just be careful with the operation you put when you brought it down.

Ah yes good good c:

Answer 22

NOW we're allowed to pull that little inverse tricky on the first part.

Answer 23

e^c = c right

Answer 24

since its an arbitrary constant?

Answer 25

Yes, very good. e^c=C Given that C must be larger than 0.

Answer 26

i get c = 5

Answer 27

but the answer says y=2(x+1)

Answer 28

Yah you should get 2, how did you get 5?

I think you made a silly mistake maybe.

Answer 29

\[\large y=Ce^{\ln(x+1)} \quad \rightarrow \quad y=C(x+1)\]
\[\large y(4)=10 \quad \rightarrow \quad 10=C(4+1)\]

Answer 30

its multiplied by c

Answer 31

how come?

Answer 32

Because of the Exponential Law that we applied, we wrote it as multiplication of bases.

Remember the e term?

Answer 33

oh i see

Answer 34

You have to be careful with problems like this, because the constant can change from addition to multiplication, depending on what you do to the problem.

Answer 35

i always make mistakes like that

Answer 36

Heh :3 practice!

Answer 37

I got it thx alot

Answer 38

np.