Solution: Solving Harmonic Oscillation Differential Equation

In summary, when solving for harmonic oscillation, it is helpful to first look at Equation 2, which has a more general solution than Equation 1. However, there may be reasoning (probably is) to make the phase angles go away.
  • #1
Runei
193
17
Hello,

When I have the differential equation

[tex]\frac{dY(x)}{dx} = -k^2 Y(x)[/tex]

The solution is of course harmonic oscillation, however, looking at various places I see the solution given as:
[tex]Y(x) = A cos(kx) + B sin(kx)[/tex]
instead of
[tex]Y(x) = A cos(kx + \phi_1) + B sin(kx + \phi_2)[/tex]

Isnt Equation 2 a more general solution than Equation 1? Or is there some reasoning (probably is) to make the phase angles go away?

Thank you.
 
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  • #2
The general solution to a second order differential equation has 2 arbitrary coefficients, so you can't have a "more general solution" than ##A \cos kx + B \sin kx##.

For your second solution, you have ##\cos(kx + \phi_1) = \cos\phi_1\cos kx - \sin\phi_1\sin kx##, and a similar expression for ##\sin(kx + \phi_2)##, so you can rewrite the whole expression as ##P\cos kx + Q\sin kx##, where ##P## and ##Q## are constants containing ##A##, ##B##, and ##\cos## and ##\sin## of ##\phi_1## and ##\phi_2##. That is the same as your first solution.

Note, ##A\cos(kx + \phi)## or ##A\sin(kx + \phi)## are both general solutions (with two arbitrary constants), and those forms are sometimes nicer to use than ##A \cos kx + B \sin kx##.
 
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  • #3
Thank you AlephZero!

As a side-note then, if I wanted to rewrite the solution in terms of complex exponentials, that the solution would be

[itex]Y(x) = A \cdot e^{ikx} + A^*\cdot e^{-ikx}[/itex]

Where the constant A this time is complex.
 
  • #4
Runei said:
Hello,

When I have the differential equation

[tex]\frac{dY(x)}{dx} = -k^2 Y(x)[/tex]

The solution is of course harmonic oscillation, however, looking at various places I see the solution given as:

Isnt Equation 2 a more general solution than Equation 1? Or is there some reasoning (probably is) to make the phase angles go away?

Thank you.
I suppose that the equation is [tex]\frac{d^2Y(x)}{dx^2} = -k^2 Y(x)[/tex]
The two equations :
[tex]Y(x) = A cos(kx) + B sin(kx)\\
Y(x) = A cos(kx + \phi_1) + B sin(kx + \phi_2)[/tex]
are equivalent :
[tex]A cos(kx + \phi_1) + B sin(kx + \phi_2) = A' cos(kx) + B' sin(kx)\\
A' =A cos(\phi_1)+B sin(\phi_2)\\
B' =-A sin(\phi_1)+B cos(\phi_2)[/tex]
 
  • #5
Runei said:
Thank you AlephZero!

As a side-note then, if I wanted to rewrite the solution in terms of complex exponentials, that the solution would be

[itex]Y(x) = A \cdot e^{ikx} + A^*\cdot e^{-ikx}[/itex]

Where the constant A this time is complex.

There is no mathematical reason why the solution has to be real, so you could just write
[itex]Y(x) = A \cdot e^{ikx} + B\cdot e^{-ikx}[/itex]
where ##A## and ##B## are complex constants.

In this case, both the real and imaginary parts of ##Y(x)## satisfy the differential equation.
You can interpret the real part of ##Y(x)## as a physical displacement, and the real part of ##dY(x)/dx## as a physical velocity, etc.
 

Related to Solution: Solving Harmonic Oscillation Differential Equation

1. What is the harmonic oscillation differential equation?

The harmonic oscillation differential equation is a second-order differential equation that describes the behavior of a system undergoing simple harmonic motion. It is written as d2x/dt2 = -kx, where x is the displacement of the system, t is time, and k is the spring constant.

2. How do you solve the harmonic oscillation differential equation?

To solve the harmonic oscillation differential equation, we use the technique of separation of variables. This involves separating the variables x and t and then integrating both sides of the equation. We then use initial conditions to determine the constants of integration and obtain the solution for x as a function of t.

3. What is the physical significance of the solution to the harmonic oscillation differential equation?

The solution to the harmonic oscillation differential equation represents the displacement of the system as a function of time. It shows how the system oscillates back and forth around its equilibrium position, with a period determined by the mass and spring constant. The solution also allows us to calculate other important quantities, such as velocity and acceleration, at any given time.

4. Can the harmonic oscillation differential equation be applied to real-world systems?

Yes, the harmonic oscillation differential equation can be applied to many real-world systems, such as a mass-spring system or a pendulum. It is a fundamental equation in classical mechanics and is used to model various physical phenomena, such as sound waves, electrical circuits, and molecular vibrations.

5. What is the relationship between the solution to the harmonic oscillation differential equation and the frequency of the oscillation?

The frequency of the oscillation is related to the solution to the harmonic oscillation differential equation by the formula f = 1/2π√(k/m), where f is the frequency, k is the spring constant, and m is the mass of the system. This means that the higher the spring constant or lower the mass, the higher the frequency of the oscillation.

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