Solving a Tricky Differential Equation Problem with Anton

  • Thread starter Hyperreality
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In summary, my friend found this problem from Antoniadi's text book on Differential Equations, and was able to solve it using L'Hospital's rule.
  • #1
Hyperreality
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My friend found this problem from Anton

Suppose that the auxiliary equation of the equation

[tex]y'' + py' + qy = 0[/tex]

has a distinct roots [tex]\mu[/tex] and [tex]m[/tex].

(a)Show that the function

[tex]g_\mu(x) = \frac{e^{\mu x} - e^{mx} }{\mu - m}[/tex]

is a solution of the differential equation

(b)Use L'Hopital's rule to show that

[tex]\lim_{\mu\rightarrow\ m} g_\mu(x) = xe^{mx}[/tex]

I tried to proof this using the D-operator method to find the roots, it doesn't seem to work. There seems to be a simpler way of doing this, but I just can't see it.

Any help is appreciated.
 
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  • #2
Ok i haven't given this much thought but try this ,
(Assuming that p and q are constant)
Standard results of polynomials,
p = mu + m
q = mu*m

Differentiate the function g(x) and substitute in the original equation to show that it satisfies the equation.

the answer to b is trivial using L'Hospital,
Differentiate numerator and denominator w.r.t mu,
its easy to see that numerator differentiates to xe^(mu*x)
and the denominator is 1.
the limit evaluates to the required one easily...

-- AI
 
  • #3
actually you don't have to use L'Hopital, just the plain definition of derivative, which is more elegant i think :P
 
  • #4
I's definitely more elegant...
The auxiliary equation reads
[tex]\lambda^2+p\lambda+q=0 [/tex]
If u chose the solution with "+" to be"µ",and the one with "-" to be "m",then its solutions verify identically the equation above,i.e.
[tex] \mu^2+p\miu+q=0;m^2+pm+q=0 [/tex]
Making the differentiations of "g" correctly and substituting into the original equation,after separating parts with [tex] \exp{\mux} [/tex] and [tex] \exp{mx} [/tex],will find exactly the 2 equations stated above,which actually will ensure you that g is a solution of th equation.
 
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  • #5
dextercioby said:
I's definitely more elegant...
The auxiliary equation reads
[tex]\lambda^2+p\lambda+q=0 [/tex]
If u chose the solution with "+" to be"µ",and the one with "-" to be "m",then its solutions verify identically the equation above,i.e.
[tex] \mu^2+p\miu+q=0;m^2+pm+q=0 [/tex]
Making the differentiations of "g" correctly and substituting into the original equation,after separating parts with [tex] \exp{\mux} [/tex] and [tex] \exp{mx} [/tex],will find exactly the 2 equations stated above,which actually will ensure you that g is a solution of th equation.

Hold your horses for a while,junior...
1.It's irrelevant "which is which",as long as they are different.
2.Why would complicate that much?The general solution to the given ODE is a linear superposition of fundamental solutions,which are [tex] \exp{\mux} [/tex] and [tex] \exp{mx} [/tex] with arbitrary (hopefully noninfinite,in this case it applies,sice "µ" and "m"are different) coefficients,call them A and B.Who stops you from chosing
[tex] A=\frac{1}{\mu-m};B=-A [/tex] or viceversa,to find your solution without making any derivatives of the solution given?? :eek:
Bonehead... :rofl: :biggrin:
 

1. What is the best approach to solving a tricky differential equation problem with Anton?

The best approach to solving a tricky differential equation problem with Anton is to start by understanding the basic concepts and techniques involved in solving differential equations. This includes knowledge of various methods such as separation of variables, substitution, and integration. It is also important to carefully analyze the given problem to identify any patterns or similarities to previous problems you have encountered.

2. How can I improve my understanding of differential equations to better solve problems with Anton?

To improve your understanding of differential equations, it is recommended to practice solving various types of problems and to seek help from a tutor or professor if needed. Reading textbooks and online resources can also provide a solid foundation for understanding the concepts and techniques involved in solving differential equations.

3. Can Anton help me with both first-order and higher-order differential equations?

Yes, Anton is equipped to handle both first-order and higher-order differential equations. However, the complexity of the problem may vary, and it is important to fully understand the techniques and methods involved in solving each type of differential equation.

4. How can I check my solution to a tricky differential equation problem with Anton?

To check your solution, you can use various online tools or software that can provide step-by-step solutions to differential equations. It is also helpful to check your answer by substituting it back into the original equation and ensuring that it satisfies the equation.

5. Are there any tips for efficiently solving a tricky differential equation problem with Anton?

Some tips for efficiently solving a tricky differential equation problem with Anton include: carefully reading and understanding the problem, breaking down the problem into smaller, more manageable parts, using appropriate techniques and methods, and checking your solution for accuracy. It is also helpful to practice regularly and seek help if needed.

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