Solve the given clairaut equation

  • Thread starter Marcin H
  • Start date
In summary, the problem is asking for the y-coordinate of the point where the x- and y-coordinates of the point (3,2) coincide.
  • #1
Marcin H
306
6

Homework Statement


Screen Shot 2016-02-11 at 1.16.04 PM.png


Homework Equations


x=-f'(t)
y=f(t)-tf'(t)

The Attempt at a Solution


Solution in picture. This is the solution to this problem, but I have no idea where the y^2=4t^6 comes from. Doin this problem I get everything up until y=2t^3 and then using x=3t^2 I solved for t and plugged it in, but that's not working. I did every other clairaut's problem like this, but I have no idea where this one goes after the y=2t^3 step.
 
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  • #2
Marcin H said:

Homework Statement


View attachment 95671

Homework Equations


x=-f'(t)
y=f(t)-tf'(t)

The Attempt at a Solution


Solution in picture. This is the solution to this problem, but I have no idea where the y^2=4t^6 comes from. Doin this problem I get everything up until y=2t^3 and then using x=3t^2 I solved for t and plugged it in, but that's not working. I did every other clairaut's problem like this, but I have no idea where this one goes after the y=2t^3 step.

You have ##x = 3t^2## and ##y=2t^3##. He is just eliminating the ##t## by noting if you cube the ##x## and sqare the ##y## both equations will have a ##t^6## so you can eliminate it easily.

It would be much easier to follow the thread if you would type the equations instead of posting images, as per forum rules.
 
  • #3
Marcin H said:

Homework Statement


View attachment 95671

Homework Equations


x=-f'(t)
y=f(t)-tf'(t)

The Attempt at a Solution


Solution in picture. This is the solution to this problem, but I have no idea where the y^2=4t^6 comes from. Doin this problem I get everything up until y=2t^3 and then using x=3t^2 I solved for t and plugged it in, but that's not working. I did every other clairaut's problem like this, but I have no idea where this one goes after the y=2t^3 step.
ohhhhhhh. I see that now. Thank you! :smile:
 

FAQ: Solve the given clairaut equation

What is a Clairaut equation?

A Clairaut equation, also known as a differential equation of the form y = x(y') + f(y'), is a mathematical equation used to describe relationships between variables in a system.

How do you solve a Clairaut equation?

To solve a Clairaut equation, you can use the method of differentiation to find the general solution. This involves finding the first derivative of the equation and substituting it back into the original equation to eliminate the second derivative. The resulting equation can then be solved for y.

What are the applications of Clairaut equations?

Clairaut equations have many applications in physics, engineering, and other fields. They are commonly used to model the motion of objects in a system, such as a pendulum or a falling object. They can also be used to describe the behavior of fluids in fluid dynamics.

Are there any special techniques for solving non-linear Clairaut equations?

Yes, there are special techniques for solving non-linear Clairaut equations, such as the method of substitution or the method of integrating factors. These techniques involve manipulating the equation to bring it into a more solvable form.

Can Clairaut equations have multiple solutions?

Yes, Clairaut equations can have multiple solutions. This is because they are second-order differential equations, meaning there are two arbitrary constants in the general solution. These constants can take on different values, resulting in different solutions to the same equation.

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