Finding Stationary Points for the Differential Equation y=(lnx)^2/x

In summary, the conversation is about finding the stationary points for the equation ## y= ((ln x)^2)/x ##. The conversation includes a solution attempt using the derivative and a request for clarification on the intention of a statement. The final output is the two stationary points, x=1 and x=e^2, and a reminder to use proper syntax for TeX/LaTex.
  • #1

chwala

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Homework Statement


Get the two stationary points for the equation ## y= ((ln x)^2)/x ##

Homework Equations

The Attempt at a Solution


i have managed to solve
##dy/dx=((2xlnx/x- (ln x)^2))/x^2 = 0,
ln x(2-ln x) = 0,
x= 1, x =e^2##
 
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  • #2
chwala said:
therefore
dy/dx=2u/u=2dy/dx=2u/u=2dy/dx=2u/u = 2 is this correct?
I think you are trying to say here, dy/du = 2v/u and it's not equal to 2.
 
  • #3
I can follow $$
{ dy\over dx} ={2x\ln x/x- (ln x)^2)\over x^2} = 0 \ \ \Leftrightarrow\ \ \ln x(2-ln x) = 0 \ \ \& \ \ x\ne 0 $$
which is satisfied for ##x= 1## and for ##x =e^2##.
But the 'then I am getting' seems a bit superfluous to me. What do you intend to show with that ?
 
  • #4
chwala said:

Homework Statement


Get the two stationary points for the equation ## y= ((ln x)^2)/x ##

Homework Equations

The Attempt at a Solution


i have managed to solve
##dy/dx=((2xlnx/x- (ln x)^2))/x^2 = 0,
ln x(2-ln x) = 0,
x= 1, x =e^2##

So, what is your question?

Anyway, please use proper syntax for TeX/LaTex: you should type "\ln ..." instead of "ln ...", because leaving out the backslash produces ugly results that are hard to read, like this (##ln x##) while using "\ln..." produces good-looking, easier-to-read results, like this (##\ln x##). BTW: the same goes for sin/arcsin, cos/arccos, tan/arctan, exp, log, max, min, lim, sinh, cosh, tanh, etc: leaving out the backslash gives ugly, hard-to-read results ##sin x##, ##arcsin x##, ##cos x##, ##arccos x##, ##tan x##, ##arctan x##, ##exp x##, ##log x##, ##max x##, ##min x##, ##lim_{x \to 0}##, ##sinh x##, etc., etc. Using the backslash produces much nicer output: ##\sin x##, ##\arcsin x##, ##\cos x##, ##\arccos x##, ##\tan x##, ##\arctan x##, ##\exp x##, ##\log x##, ##\max x##, ##\min x##, ##\lim_{x \to 0}##, ##\sinh x##, etc.
 
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  • #5
I a m sorry the question was to find the co ordinates of the stationary point for the given function ## y= f(x)##
 
  • #6
That is the problem statement. What Ray means is: what question do you want your helpers to answer ?
 
  • #7
I have answered the question already sorry
 

What are differential equations?

Differential equations are mathematical equations that involve an unknown function and its derivatives. They are used to model a wide range of physical phenomena in fields such as physics, engineering, and economics.

What is the difference between ordinary and partial differential equations?

Ordinary differential equations involve only one independent variable, while partial differential equations involve multiple independent variables. Ordinary differential equations can be solved using techniques such as separation of variables, while partial differential equations require more advanced methods.

Why are differential equations important?

Differential equations are important because they allow us to model and predict the behavior of complex systems in various fields of science and engineering. They are also used to develop mathematical models that can be used to make decisions and solve real-world problems.

What are some real-world applications of differential equations?

Differential equations are used in a variety of real-world applications, such as modeling population growth, predicting the spread of diseases, analyzing stock market trends, and designing control systems for engineering processes.

How do you solve a differential equation?

There are various methods for solving differential equations, depending on the type of equation and its complexity. Some common techniques include separation of variables, substitution, and using special functions such as the Laplace transform. In some cases, numerical methods may also be used to approximate solutions.

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