Solve Diferential Equation: d2u/(dθ)^2+u=0 → u=cos(θ-θ0)

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In summary: Yes, thank you, but how can I get that resultIn summary, you can differentiate it twice, replace it with the equation, and get to zero.
  • #1
alpha25
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How can I demostrate that a solution of d2u/(dθ)^2+u=0 is u=cos(θ-θ0)

Thanks
 
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  • #2
alpha25 said:
How can I demostrate that a solution of d2u/(dθ)^2+u=0 is u=cos(θ-θ0)

Thanks

Differentiate it twice, add it to the result, and see if you get zero.
 
  • #3
Yes, thanks, but how can I get that result
 
  • #4
Show us what happens when you try it.
 
  • #5
I already differentiate it twice and then I replace it to the equation and I get to zero, I know that cos(θ-θ0) is a solution, but I don t know how to get that solution.
When I solve the equation I reach other result more complicated with imaginary terms etc...
 
  • #6
To save typing I will use ##x## instead of ##\theta## for the independent variable. When you solved ##u''+u=0## you probably got solutions like ##e^{ix}## and ##e^{-ix}##, so the general solution is ##u = Ae^{ix}+Be^{-ix}##. Using the Euler formulas you can write this equivalently as ##u = C\cos x + D\sin x##. A solution of the form ##\cos(x-x_0)## can be written using the addition formula as ##\cos x \cos x_0 - \sin x \sin x_0##. You can get that from the previous form by letting ##C=\cos x_0,~D= -\sin x_0##.

You can read about constant coefficient DE's many places on the internet. One such place is:
http://www.cliffsnotes.com/math/differential-equations/second-order-equations/constant-coefficients
 
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  • #7
D is imaginary?
 
  • #8
The constants can be imaginary or complex. But there is a theorem that if the coefficients of the DE are real and the boundary conditions are real, the constants C and D will be real in the {sine,cosine} expression. If you leave the solution in the complex exponential form, the constants A and B will come out complex. So for real coefficients and real boundary conditions, you really just make it complicated if you leave it in the complex exponential form. Use the {sine,cosine} form.
 
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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 often used to model physical phenomena and can be solved to find the function that satisfies the equation.

2. How do you solve a differential equation?

The method for solving a differential equation depends on its type and order. In general, the goal is to find a solution that satisfies the equation and any given initial conditions. This can be done analytically, using techniques such as separation of variables, or numerically, using computational methods.

3. What is the order of this differential equation?

The order of a differential equation is the highest derivative present in the equation. In this case, the highest derivative is d2u/(dθ)^2, so the order is 2.

4. What is the particular solution for this differential equation?

The particular solution for this differential equation is u=cos(θ-θ0). This solution has been obtained by solving the equation d2u/(dθ)^2+u=0, which represents a simple harmonic motion.

5. What is the significance of θ0 in the solution?

θ0 represents the initial phase of the solution. It determines the starting point or position of the function u=cos(θ-θ0) on the θ-axis. Changing the value of θ0 will result in a different starting position, but the overall shape and behavior of the function will remain the same.

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