Question

AP Calculus: Slope Fields/ Separate Variables
Find the equation of the curve, given the derivative and the indicated point on the curve.
dy/dx=lnx/x

Answer 1

What do you find confusing about how to solve the differential equation? Have you already learned the method of separation of variables?

Answer 2

kinda but i was absent that day when we learned it in class :\

Answer 3

The first step in any differential equation problem is to determine whether you can separate variables. That is to say, can you move all the terms with "dy" and "y" to one side, and "dx" and "x" to the other?

Answer 4

yes :)

Answer 5

Now do so; in this case, you can treat "dx" as its own term and multiply both sides by it. Then you've got\[dy = \frac{ \ln{x} dx }{ x }\] Now integrate both sides and you're virtually done!

Answer 6

so far i have \[\int\limits_{}^{} dy = \int\limits_{}^{} \frac{ lnx }{ x }dx\]

Answer 7

i dont know how to integrate \[\int\limits_{}^{} \frac{ lnx }{ x } dx\] though :\

Answer 8

integration by parts?

Answer 9

would i use substituion?

Answer 10

There's no harm in trying a few different methods! This one's simple enough that it shouldn't take too much time.

Answer 11

so would i use the product rule?

Answer 12

I think that'd be a bit tedious, as integration by parts problems usually are. Try a substitution.

Answer 13

so then i use substitution for this?

Answer 14

so you split it up to\[\int\limits_{}^{} \ln x \times \frac{ 1 }{ x } dx\]

Answer 15

It's a good rule of thumb to try substitution when you see fractions in the integrand. The problem simplifies very quickly with a particular substitution.

Answer 16

ok so when i get \[\int\limits_{}^{} \ln x \times \frac{ 1 }{ x } dx \] what do i next to integrate it?

Answer 17

Have you tried a substitution? Think of the most obvious one that might work.

Answer 18

yeah i have but idk how to get x

Answer 19

so let \[u=\ln x\] right?

Answer 20

You tell me if it's right.

Answer 21

Carry it out. :)

Answer 22

ok but how would i find x

Answer 23

is x \[\frac{ u }{ \ln }= x\]

Answer 24

The goal of substitution is to eliminate your original variable, in this case it was x. Differentiate your substitution to get \[du = \frac{1}{x} dx\] and you can rearrange it to get dx. Then, put that dx into your original integral and every term with x in it will have disappeared and you will have a simple integral in u and du.

Answer 25

so u isn't \[u=\ln x\]

Answer 26

It is, but you have to differentiate that equation you've made.

Answer 27

but what would i substitue for x?

Answer 28

Once you differentiate your substitution, you get the statement I said above, and you acquire \[dx = x du\] which you can put into your original integral. The x's will cancel out and your ln(x) turns into a u, as you already said it would.

Answer 29

You end up with \[\int\limits u \ du\] which I think you know how to handle.

Answer 30

i'm still confused

Answer 31

for substitution u have to find u, x, and du. \[u = \ln x\]

Answer 32

Yes. Differentiate it, and what do you have?

Answer 33

\[u=lnx\] would you divide ln to get x by itself?

Answer 34

No, ln(x) is a function of x. You can't divide by ln. What is the derivative of ln(x)?

Answer 35

\[\frac{ 1 }{ x }\]

Answer 36

So you have du/dx = 1/x. Now you want to find what dx is, in order to replace it in your original integral.

Answer 37

so i divide by du so i would get \[\frac{ 1 }{ dx }= \frac{ 1 }{ x \times du }\]

Answer 38

Yes, and you can flip both sides to get dx = x*du. Now, you have dx = x*du, and you have u = ln(x). Put those both into your integral from before.

Answer 39

are you sure substitution is the only way for this problem?

Answer 40

cause i don't really like using substituion. My Calculus teacher said that should be our last resolution in dealing with integrals..

Answer 41

I'm sure it's not the only way, but it certainly is the fastest and simplest in most cases.

Answer 42

when you explain your way of method of substitution you confuse me :\

Answer 43

when we do substitution the equation usually has a square root

Answer 44

can you show me another method on how to get the integral of \[\int\limits_{}^{} \frac{ lnx }{ x }dx\] cause i'm pretty sure my teacher doesn't like seeing us do substituion with these types of problems unless it involves a square root in them.

Answer 45

Your teacher shouldn't have a preference either way, it's just as valid a method as any other, and you're less likely to make mistakes with this substitution because integration by parts tends to be easy to lose track of.

Answer 46

well can you show me the product rule for this one or anything besides substitution? I have a test on this tomorrow :\

Answer 47

Using product rule, choosing u = ln(x) and dv/dx = 1/x, we have \[\int\limits \ln(x) \frac{ 1 }{ x } dx = \ln(x) \ln(x) - \int\limits \ln(x) \frac{1}{x} dx\]
Then, add the integral on the right-hand side to the left hand side, and divide the whole expression by two to get\[y = \frac{\ln^2(x)}{2} + c\]

Answer 48

A great rule I was taught to determine which term to set equal to u is "LIPET" - Logarithms, inverse-trigonometric functions, polynomials, exponentials, and trigonometric functions. Whichever comes first in the lineup will be the one you should pick for your "u", and it will make your integral easier to solve.

Answer 49

yeah i still dont get it haha :P