Strange way of solving a linear 2nd order DE

In summary, the conversation discusses a given differential equation and the attempt to solve it using a separable DE method. The solution was eventually found to be \Phi \propto \eta^{-5} + constant. The conversation also clarifies the use of the independent variable and the differentiation method.
  • #1
Alexrey
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0

Homework Statement


I was given a DE of the form: [tex]\Phi^{''}+(6/\eta)\Phi^{'}=0[/tex] where the next step was given as [tex]\Phi^{'} \propto \eta^{-6}[/tex] where the answer came out to be [tex]\Phi \propto \eta^{-5} + constant[/tex]

The Attempt at a Solution


My attempt was to set [tex]\Phi^{'}=x[/tex] where I would then get [tex]x^{'}=-(6/\eta)x[/tex] and then solve for a seperable DE, but my answer was incorrect. Any help would be appreciated, thanks guys.

Any help would be appreciated, thanks guys.
 
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  • #2
Alexrey said:
[tex]x^{'}=-(6/\eta)x[/tex]
To clarify, η is the independent variable, and differentiation is wrt that? If so [tex]dx/x=-6d\eta/\eta[/tex] yes? Isn't it straightforward from there?
 
  • #3
About 5 minutes after posting this I figured it out. :/ Thanks for replying though, I appreciate it, that's exactly what I got. Cheers!
 

FAQ: Strange way of solving a linear 2nd order DE

1. What is a linear 2nd order DE?

A linear 2nd order DE (differential equation) is a mathematical equation that describes the relationship between a function, its derivatives, and the independent variable, where the highest derivative is raised to the second power and the function and its derivatives are multiplied by constant coefficients. It can be written in the form: y'' + p(x)y' + q(x)y = r(x).

2. What makes the way of solving a linear 2nd order DE "strange"?

The "strange" way of solving a linear 2nd order DE refers to a specific method called the "method of undetermined coefficients". It involves guessing a particular solution based on the form of the non-homogeneous part of the equation and then using that solution to find the general solution. This method is considered "strange" because it relies on trial and error rather than a systematic approach.

3. Can the "strange" way of solving a linear 2nd order DE be used for all types of DEs?

No, the method of undetermined coefficients can only be used for linear 2nd order DEs with constant coefficients and non-homogeneous terms that can be expressed as a polynomial, exponential, sine, or cosine function.

4. Are there other methods for solving linear 2nd order DEs?

Yes, there are other methods such as the method of variation of parameters, the Laplace transform method, and the method of power series. These methods are more systematic and do not rely on guessing a particular solution.

5. What are some real-world applications of solving linear 2nd order DEs?

Linear 2nd order DEs are commonly used in physics, engineering, and economics to model various systems and phenomena. For example, they can be used to describe the motion of a pendulum, the decay of radioactive materials, and the growth of a population. They are also used in signal processing and control theory to analyze and optimize systems.

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