What Is the Constant Solution to the Given Differential Equation?

[ssb]

Homework Statement

Consider the following differential equation
y' = t*ln(y^(2t))+t^2

There is one constant solution to this problem. Find it
Basically I have to get y' and all other y on the left, and leave all the t's on the right.

Homework Equations

The Attempt at a Solution

Its just algebra but I can't remember all the rules with e and ln.

 

[ChaoticLlama]

remember:

ln(a^b) = b*ln(a)

what does that allow you to do?

 

[Dick]

Luckily, you don't have to solve this equation to answer the question. If y is a constant solution what can you say about y'?

 

[ssb]

ChaoticLlama Thanks for that tip. I had forgotten that one. I went on using your tip and I came across this when I separated the equations:

1/ln(y) dy = 2t^2 + t dt

First off I am not sure how to integrate 1/ln(y)

-----------------------------

Dick, if y' has a constant solution, then y at a given point has a constant solution? are you suggesting that I don't have to solve one side for y and I can just plug in the numbers or something? confused

Thanks for the help guys!

 

[Dick]

Well, you didn't separate the equation correctly. So I wouldn't worry about integrating 1/ln(y). I was trying to get you to realize that if y is a constant solution then y'=0. The trouble with this is that the equation as written doesn't seem to have any constant solutions. As you sure you transcribed it correctly??

 

[Data]

Are you sure that you typed out the DE correctly? Is it

$$y^\prime = t\ln \left(y^{2t}\right) + t, \ \mbox{or } \ y^\prime = t(\ln \left(y^{2t}\right) + t)?$$

The first one (which is what's in your first post) does not have a constant solution at all. The second one does.

With regard to Dick's suggestion, think about it: If y is constant, then what's dy/dt?

 

[ssb]

Well, you didn't separate the equation correctly. So I wouldn't worry about integrating 1/ln(y). I was trying to get you to realize that if y is a constant solution then y'=0. The trouble with this is that the equation as written doesn't seem to have any constant solutions. As you sure you transcribed it correctly??

jesus your right. I am such an idiot. i even double checked it and triple checked it too... the last T has is squared

y' = t*ln(y^(2t))+t^2

I feel so stupid

 

[Data]

Okay, that's the second one I posted up there. So use Dick's suggestion and solve the resulting eq.

 

[Dick]

jesus your right. I am such an idiot. i even double checked it and triple checked it too... the last T has is squared

y' = t*ln(y^(2t))+t^2

I feel so stupid

Happens to the best of us. Now it has a constant solution. Can you find it?

 

[ssb]

$$\ y^\prime = t \ln \left(y^{2t}\right) + t^2$$

$$\ y^\prime = t * (2t) \ln \left(y) +t^2$$

$$\ 1/\ln \left(y) y^\prime = t (2t) + t^2$$

$$\ 1/\ln \left(y) y^\prime = 3t^2$$

$$\ \int 1 / ln (y) (y^\prime) = \int 3t^2$$

$$\ \int 1 / ln (y) (dy) = \int 3t^2 dt$$

$$\ ? = t^3 + C$$With your clues that you were giving me on the fact that y was constant, i thought that you meant that

$$\ 0 = t^3 + C$$

but I think I am going the wrong way with this

 

[Data]

I'll repeat again: If y is constant, then what's y' = dy/dt (remember, intuitively dy/dt is the rate of change of y with respect to t)?

 

[ssb]

I'll repeat again: If y is constant, then what's y' = dy/dt (remember, intuitively dy/dt is the rate of change of y with respect to t)?

It would be zero. Give me a minute to think about this one... i know the significance should probably be smacking me in the face right about now. All I can think is that the derivative of a constant is zero so part of my equation should equal zero?

In post 10 I am trying to show my steps because I am just not getting it and I am sorry. Calculus is really killing me this quarter.

 

[HallsofIvy]

It would be zero. Give me a minute to think about this one... i know the significance should probably be smacking me in the face right about now. All I can think is that the derivative of a constant is zero so part of my equation should equal zero?

Since your equation is y'= f(x) the whole equation should be 0! If y(t) is a constant function then
y'(t)= t*ln(y^(2t))+t^2= 0. Now solve for y (remember that it should NOT depend on t because it is a constant). (Of course, the fact that $ln(y^{2t})= 2t ln(y)$ is useful!)

 

[Dick]

$$\ y^\prime = t \ln \left(y^{2t}\right) + t^2$$

$$\ y^\prime = t * (2t) \ln \left(y) +t^2$$

$$\ 1/\ln \left(y) y^\prime = t (2t) + t^2$$

$$\ 1/\ln \left(y) y^\prime = 3t^2$$

$$\ \int 1 / ln (y) (y^\prime) = \int 3t^2$$

$$\ \int 1 / ln (y) (dy) = \int 3t^2 dt$$

$$\ ? = t^3 + C$$


With your clues that you were giving me on the fact that y was constant, i thought that you meant that

$$\ 0 = t^3 + C$$

but I think I am going the wrong way with this

You can't separate the variables in that equation unless you make an algebraic mistake. Which you did. Then other mistakes ensue. I don't know why you won't believe the advice you've been given repeatedly that you don't have to solve a differential equation.

 

[ssb]

You can't separate the variables in that equation unless you make an algebraic mistake. Which you did. Then other mistakes ensue. I don't know why you won't believe the advice you've been given repeatedly that you don't have to solve a differential equation.

Im sorry. I just don't understand the advice i guess. I have yet to successfully solve one of these problems for my class. I am used to single variable calculus and this is the first time I've ever seen 2 variables in a math problem such as this. its the first week of class and the teacher expects us to already know how to do this kind of stuff. Last quarter we didnt cover these in any of our topics.

Maybe do you know of a good website that shows example problems worked out step by step so I can see what's expected to be done? I've searched google and wikipedia and cannot find any good examples.

All i know is that he's looking for a "Constant solution" to this problem. I don't know if the solution will contain just numbers, y, t, or a combination of the 3. To be quite honest I am not even sure what form to put such an answer in. sorry.

The equation is
$$y'=t\,\ln(y^{2t})+t^2$$
and the question states that it is separable. Can someone help me separate it and let me see if I can get it from there? Thanks

 

[d_leet]

All i know is that he's looking for a "Constant solution" to this problem. I don't know if the solution will contain just numbers, y, t, or a combination of the 3. To be quite honest I am not even sure what form to put such an answer in. sorry.

What does constant mean? A constant solution should contain only constants.

The equation is
$$y'=t\,\ln(y^{2t})+t^2$$
and the question states that it is separable. Can someone help me separate it and let me see if I can get it from there? Thanks

It is separable, but since you are looking for a constant solution, as numerous others have pointed out, this is not the way to go about this problem.

 

[HallsofIvy]

I told you before, if y is a constant then its derivative is 0 so you have y'(t)= t*ln(y^(2t))+t^2= 0. Solve the equation t*ln(y^(2t))+t^2= 0 for y!

 

[HallsofIvy]

I told you before: if y is a CONSTANT solution, then $y'(t)= t ln(y^{2t})+t^2= 2t^2 ln(y)+ t^2= 0$. Solve that for y.