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

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

Thanks

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.

Yes, thanks, but how can I get that result

Show us what happens when you try it.

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...

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

1 person
D is imaginary?

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.

1 person

## 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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