How do I solve this differential equation?

In summary: However, it's pointless to do so if you made a mistake in the first place.You've calculated the derivative of f with respect to y' the wrong way. You should have differentiated with respect to t instead.
  • #1
LFCFAN
22
0

Homework Statement


Solve the following differential equation:


Homework Equations



2 y (y'')^2 + 2 y y''' y' -2 (y')^2 y'' = - (y')^2



The Attempt at a Solution



I don't know if the following is useful, but if you divide both sides by y^2, the LHS of the above becomes:

(2 y (y'')^2 + 2 y y''' y' -2 (y')^2 y'')/y^2 = ( (2 y' y'')/y )'

Which means the equation to solve is;

( (2 y' y'')/y )' = - (y')^2
 
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  • #2
If you divide both sides of your original DE by y^2, it's not obvious how that turns into the equations in section 3. above.
 
  • #3
@SteamKing

Reverse quotient rule. All I know is that what I wrote is definitely correct. I just have no idea how to solve it.
 
  • #4
LFCFAN said:
@SteamKing

Reverse quotient rule. All I know is that what I wrote is definitely correct. I just have no idea how to solve it.

Steam king is right, you didn't divide through properly at all.

If this is your equation : ##2(y'')^2y + 2 y'''y'y - 2y''(y')^2 = -(y')^2##

Then dividing through by ##y^2## would yield : ##\frac{2(y'')^2}{y} + \frac{2y'''y'}{y} + 2y'' {(\frac{y'}{y})}^2 = -{(\frac{y'}{y})}^2##
 
  • #5
Why are you guys so fixated on that line??

That's not even the crux of the question I'm asking!

lol


I obtained this differential equation from applying the Euler-Lagrange equation to the following function:

f(y,y') = (y')^2/y


That's how I know my reverse quotient rule is correct.



So how do I solve the original ODE?



Thanks!
 
  • #6
Because it's pointless to answer the question if you messed up in the first place, and despite your claim that you know it's right, there's an obvious algebra mistake in the third line. It's probably just a typo, but you should recognize why the helpers want to make sure you aren't wasting their time with the wrong question.
 
  • #7
6 replies and all of them useless.

Fixated on something frivolous. I gave you guys the correct differential equation in the first line anyway, so there's absolutely no problem imo.
 
  • #8
Well, half of those six replies are from you, so... nyuk, nyuk, nyuk...

How did you manage to get that differential equation from ##f(y,y') = \frac{y'^2}{y}##? In particular, how'd you manage to end up with a third derivative?
 
  • #9
By plugging (y')^2/y into the Euler-Lagrange Equation
 
  • #10
You must have done it wrong.
 
  • #11
I've done it wrong because you can't solve the equation... right...
 
  • #12
partial derivative of (y')^2/y wrt y = - (y')^2/y^2

partial derivative of (y')^2/y wrt y' = (2 y' y'')/y

derivative of (2 y' y'')/y wrt x = (2 y (y'')^2 + 2 y y''' y' -2 (y')^2 y'')/y^2 finsihed. everything I've done is correct
 
  • #13
Are you claiming you calculated
$$\frac{d}{dt} \left(\frac{\partial f}{\partial y'}\right) = \frac{\partial f}{\partial y}?$$ Because if you are, you did it wrong.
 
Last edited:
  • #14
Yes that's the equation, just swop t for x.How have I done it wrong? I cross-checked with peers...
 
  • #15
Your mistake is when you took the partial with respect to y'. It should be
$$\frac{\partial f}{\partial y'} = \frac{2y'}{y}.$$ You seem to have applied the chain rule when you shouldn't have. You're not differentiating with respect to t.

You'll end up with a differential equation that you'll easily be able to solve.
 
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1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is used to model many real-world phenomena in fields such as physics, engineering, and economics.

2. How do I know if a differential equation is solvable?

There is no general rule for determining if a differential equation is solvable. However, there are certain techniques and methods that can be used for specific types of differential equations. It is important to analyze the form and properties of the equation to determine the appropriate method for solving it.

3. What are the different methods for solving differential equations?

Some common methods for solving differential equations include separation of variables, substitution, integrating factors, and power series. The choice of method depends on the form and complexity of the equation, as well as the initial conditions and boundary conditions.

4. Can differential equations be solved analytically?

Not all differential equations can be solved analytically, meaning that an explicit formula for the solution cannot be found. In some cases, numerical methods must be used to approximate the solution. However, many simple and special types of differential equations can be solved analytically.

5. How do I check if my solution to a differential equation is correct?

To check the correctness of a solution to a differential equation, you can substitute the solution into the original equation and see if it satisfies the equation. Additionally, you can check if the solution satisfies any initial or boundary conditions given in the problem.

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