Second Order Linear Differential Equation

In summary: If you don't, they are related to your problem.On a side note, it would be much simpler to write the equation in terms of the letters that are usually used, rather than Greek letters. Translated to x and y, your equation looks like this:1/x * d/dx(x dy/dx) - β2y = 0
  • #1
Pawnag3
12
0
Hey, I'm not sure how to even approach this problem. It's not a simple ODE.

Basically, I want to find the solution for Θ in terms of ε. The equation is
[itex]\frac{1}{ε}*\frac{d}{dε}*(ε*\frac{dΘ}{dε})-β^{2}Θ=0[/itex]

I tried to move the B^2 to the other side and I wasn't able to solve it that way. I can't solve it like a normal second order ODE because it has ε in front.

Thanks for your help in advance!
 
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  • #2
Can you define the notation a bit? Are you solving for Θ? Is Θ a function of ε -> Θ(ε)? Is β just a constant? are there any initial conditions or you just need a general solution?
 
  • #3
Sorry! I want to solve for Θ, and Θ is a function of ε, β is just a constant. I just want a general solution please.
 
  • #4
Pawnag3 said:
Hey, I'm not sure how to even approach this problem. It's not a simple ODE.

Basically, I want to find the solution for Θ in terms of ε. The equation is
[itex]\frac{1}{ε}*\frac{d}{dε}*(ε*\frac{dΘ}{dε})-β^{2}Θ=0[/itex]
Part of your notation makes no sense. The equation above should not have d/dε "times" something. It means to take the derivative with respect to ε of (ε dθ/dε). You'll need to use the product rule to simplify this part.

Once you do this, you'll have a second order DE to solve.
Pawnag3 said:
I tried to move the B^2 to the other side and I wasn't able to solve it that way. I can't solve it like a normal second order ODE because it has ε in front.

Thanks for your help in advance!

On a side note, it would be much simpler to write the equation in terms of the letters that are usually used, rather than Greek letters. Translated to x and y, your equation looks like this:
1/x * d/dx(x dy/dx) - β2y = 0
 
  • #5
Pawnag3 said:
Hey, I'm not sure how to even approach this problem. It's not a simple ODE.

Basically, I want to find the solution for Θ in terms of ε. The equation is
[itex]\frac{1}{ε}*\frac{d}{dε}*(ε*\frac{dΘ}{dε})-β^{2}Θ=0[/itex]

I tried to move the B^2 to the other side and I wasn't able to solve it that way. I can't solve it like a normal second order ODE because it has ε in front.

Thanks for your help in advance!


What is stopping you from multiplying through by ε, so it will not have ε "in front"? Of course, you need ε ≠ 0, but you had to have that anyway, since you were initially dividing by ε.

BTW: do you know about Bessel functions and Bessel's differential equation?
 

Related to Second Order Linear Differential Equation

What is a Second Order Linear Differential Equation?

A Second Order Linear Differential Equation is a mathematical equation that describes the relationship between a function and its derivatives up to the second order. It follows the form y'' + p(x)y' + q(x)y = g(x), where y' represents the first derivative of y with respect to x and y'' represents the second derivative.

What are the key components of a Second Order Linear Differential Equation?

The key components of a Second Order Linear Differential Equation are the dependent variable (y), the independent variable (x), the coefficients (p and q), and the driving force or source term (g). These components can be used to solve for the unknown function y(x).

What is the order of a Second Order Linear Differential Equation?

The order of a Second Order Linear Differential Equation is 2, which refers to the highest derivative present in the equation. In this case, it is the second derivative (y'').

What are the different types of solutions to a Second Order Linear Differential Equation?

The solutions to a Second Order Linear Differential Equation can be classified as either homogeneous or non-homogeneous. A homogeneous solution only contains the dependent variable and its derivatives, while a non-homogeneous solution contains an additional term for the driving force (g). These solutions can also be further categorized as either real or complex depending on the nature of the coefficients.

What are some applications of Second Order Linear Differential Equations in science?

Second Order Linear Differential Equations have various applications in science, particularly in physics and engineering. They can be used to model oscillatory systems such as pendulums and springs, as well as describe the behavior of electric circuits and mechanical vibrations. They are also commonly used in the study of fluid dynamics and heat transfer.

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