Conditions of the Euler Equation

In summary, the Euler Equation is a mathematical equation that describes the relationship between pressure, velocity, and density of a fluid in motion. It assumes the fluid is inviscid, incompressible, and steady. It is derived from Newton's Second Law of Motion and has applications in aerodynamics, hydrodynamics, and vehicle design. However, it has limitations such as not accounting for viscosity, turbulent flows, or compressible fluids, which are addressed by other equations.
  • #1
jonneh
5
0
Hi everyone :D

This is my problem:
Find conditions on [tex]\alpha[/tex] and [tex]\beta[/tex] in the Euler equation x[tex]^{2}[/tex]y'' + [tex]\alpha[/tex]xy' + [tex]\beta[/tex]y = 0 such that:

a) All solutions approach zero as x [tex]\rightarrow[/tex] 0
b) All solutions are bounded as x [tex]\rightarrow[/tex] 0
c) All solutions approach zero as x [tex]\rightarrow\infty[/tex]

I don't really know where to start with this, actually, I have no clue where to start.

Also, what does it really mean for a solution to be bounded? I've been scouring some textbooks for a simple explanation but I can't seem to find it. Does it just mean that the function is constrained to some region?

Any help would be greatly appreciated :D
 
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  • #2
jonneh said:
Hi everyone :D

This is my problem:
Find conditions on [tex]\alpha[/tex] and [tex]\beta[/tex] in the Euler equation x[tex]^{2}[/tex]y'' + [tex]\alpha[/tex]xy' + [tex]\beta[/tex]y = 0 such that:

a) All solutions approach zero as x [tex]\rightarrow[/tex] 0
b) All solutions are bounded as x [tex]\rightarrow[/tex] 0
c) All solutions approach zero as x [tex]\rightarrow\infty[/tex]

I don't really know where to start with this, actually, I have no clue where to start.
Do you know how to find the general solution to an Euler equation? There are two standard ways. One, the simpler, is to try "y= xr" and get a polynomial equation for r. The other, more complicated, is to make the substitution u= ln(x) which converts the equation into one with constant coefficients.

Also, what does it really mean for a solution to be bounded? I've been scouring some textbooks for a simple explanation but I can't seem to find it. Does it just mean that the function is constrained to some region?
It means that the function value has some upper and lower bounds. I would not say "the function is constrained to some region" because that would mean x can only be between two limits. If a function is bounded, then its values do not go to plus or minus infinity. In particular a 'power of x', xr, is bounded as x goes to 0 if and only if r is not negative.

Any help would be greatly appreciated :D
 
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  • #3
Thank you HallsofIvy for your help!

The polynomial equation for r that you get when you plug in y = x[tex]^{r}[/tex] and try to solve the differential equation is:

r[tex]_{1}[/tex], r[tex]_{2}[/tex] = [tex]\frac{-(\alpha-1) \pm \sqrt{(\alpha-1)^{2}-4\beta}}{2}[/tex]

where r1 and r2 can be real and different, real and equal, or complex conjugates.

Am I supposed to use limits or something to show what is asked in the problem statement?

Also, would it be fair to say that (a) all solutions approach zero as x[tex]\rightarrow[/tex] 0 and (b) all solutions are bounded as x[tex]\rightarrow[/tex] 0 are roughly the same thing?
 
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  • #4
a implies b, but not the other way around, I believe. So, I suppose if you don't need the weakest possible conditions on alpha and beta, you could just give the same answer, though that seems like a bit of a copout.

But I completely suck at differential equations, so I should probably stay out of this.
 
  • #5
jonneh said:
Thank you HallsofIvy for your help!

The polynomial equation for r that you get when you plug in y = x[tex]^{r}[/tex] and try to solve the differential equation is:

r[tex]_{1}[/tex], r[tex]_{2}[/tex] = [tex]\frac{-(\alpha-1) \pm \sqrt{(\alpha-1)^{2}-4\beta}}{2}[/tex]

where r1 and r2 can be real and different, real and equal, or complex conjugates.

Am I supposed to use limits or something to show what is asked in the problem statement?
Basically, it is a matter of looking at the sign of the powers of x. If all powers of x have positive sign, the solution goes to 0 as x goes to 0 but goes to infinity as x goes to infinity. If all powers of x have negative sign, the solution goes to infinity as as x goes to 0 but goes to 0 as x goes to positive infinity. I'll let you think about what happens if one root is positive and the other negative.

Also, would it be fair to say that (a) all solutions approach zero as x[tex]\rightarrow[/tex] 0 and (b) all solutions are bounded as x[tex]\rightarrow[/tex] 0 are roughly the same thing?
For powers of x, yes. More generally however, the function cos(x) is bounded for all x and so as x goes to 0 but does not approach 0 there.
 

Related to Conditions of the Euler Equation

What is the Euler Equation?

The Euler Equation is a mathematical equation that describes the relationship between the pressure, velocity, and density of a fluid in motion. It is used to model and understand the behavior of fluids, such as air or water, in various applications including aerodynamics and fluid mechanics.

What are the conditions of the Euler Equation?

The conditions of the Euler Equation include the assumption that the fluid is inviscid, meaning it has no internal friction or viscosity. Additionally, it assumes that the fluid is incompressible, meaning its density remains constant. Finally, it assumes that the flow is steady, meaning it does not change over time.

How is the Euler Equation derived?

The Euler Equation is derived from Newton's Second Law of Motion, which states that the sum of all forces acting on an object is equal to its mass multiplied by its acceleration. This law is applied to a small volume of fluid, and using various mathematical techniques, the Euler Equation is obtained.

What are the applications of the Euler Equation?

The Euler Equation has several applications in various fields of science and engineering. It is used to study and understand the behavior of fluids in aerodynamics, hydrodynamics, and other areas of fluid mechanics. It is also used in designing and optimizing aircraft, ships, and other vehicles that move through fluids.

What are some limitations of the Euler Equation?

While the Euler Equation is a useful tool for understanding fluid flow, it has some limitations. It does not account for the effects of viscosity, which can significantly impact the behavior of fluids. It also does not consider turbulent flows or compressible fluids, which may be important in certain applications. Other equations, such as the Navier-Stokes equations, are used to address these limitations.

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