Question
How do I solve the differential equation for a simple harmonic oscillator f(x) = -kx?
Answers
GPT-4o mini
To solve the differential equation for a simple harmonic oscillator, we first need to clarify the equation you're working with. The equation you provided, \( f(x) = -kx \), is actually the force acting on a mass in simple harmonic motion, where \( k \) is the stiffness constant of the spring and \( x \) is the displacement from the equilibrium position.
The dynamics of a simple harmonic oscillator can be described by Newton's second law of motion:
\[
F = ma
\]
Here, the force \( F \) is given by Hooke's law:
\[
F = -kx
\]
Setting these equal, we can express the acceleration \( a \) in terms of displacement \( x \):
\[
ma = -kx
\]
Substituting \( a \) with the second derivative of displacement \( x \):
\[
m \frac{d^2x}{dt^2} = -kx
\]
We can rearrange this equation to obtain the standard form of the simple harmonic oscillator differential equation:
\[
\frac{d^2x}{dt^2} + \frac{k}{m} x = 0
\]
Let \( \omega^2 = \frac{k}{m} \). The equation then simplifies to:
\[
\frac{d^2x}{dt^2} + \omega^2 x = 0
\]
### Solution to the Differential Equation
This is a second-order linear ordinary differential equation with constant coefficients. The general solution can be found by assuming a solution of the form:
\[
x(t) = C_1 \cos(\omega t) + C_2 \sin(\omega t)
\]
where \( C_1 \) and \( C_2 \) are constants determined by initial conditions.
### Determining Constants
To find \( C_1 \) and \( C_2 \), we need initial conditions, such as:
1. The initial displacement \( x(0) = x_0 \)
2. The initial velocity \( \frac{dx}{dt}(0) = v_0 \)
Applying the initial conditions to the general solution:
1. At \( t = 0 \):
\[
x(0) = C_1 \cos(0) + C_2 \sin(0) = C_1 = x_0
\]
So, \( C_1 = x_0 \).
2. The velocity is given by the derivative of \( x(t) \):
\[
\frac{dx}{dt} = -C_1 \omega \sin(\omega t) + C_2 \omega \cos(\omega t)
\]
Evaluating at \( t = 0 \):
\[
\frac{dx}{dt}(0) = -C_1 \omega \sin(0) + C_2 \omega \cos(0) = C_2 \omega = v_0
\]
Thus, \( C_2 = \frac{v_0}{\omega} \).
### Final Solution
Putting it all together, the solution to the differential equation is:
\[
x(t) = x_0 \cos(\omega t) + \frac{v_0}{\omega} \sin(\omega t)
\]
This represents the motion of a simple harmonic oscillator with initial displacement \( x_0 \) and initial velocity \( v_0 \).
The dynamics of a simple harmonic oscillator can be described by Newton's second law of motion:
\[
F = ma
\]
Here, the force \( F \) is given by Hooke's law:
\[
F = -kx
\]
Setting these equal, we can express the acceleration \( a \) in terms of displacement \( x \):
\[
ma = -kx
\]
Substituting \( a \) with the second derivative of displacement \( x \):
\[
m \frac{d^2x}{dt^2} = -kx
\]
We can rearrange this equation to obtain the standard form of the simple harmonic oscillator differential equation:
\[
\frac{d^2x}{dt^2} + \frac{k}{m} x = 0
\]
Let \( \omega^2 = \frac{k}{m} \). The equation then simplifies to:
\[
\frac{d^2x}{dt^2} + \omega^2 x = 0
\]
### Solution to the Differential Equation
This is a second-order linear ordinary differential equation with constant coefficients. The general solution can be found by assuming a solution of the form:
\[
x(t) = C_1 \cos(\omega t) + C_2 \sin(\omega t)
\]
where \( C_1 \) and \( C_2 \) are constants determined by initial conditions.
### Determining Constants
To find \( C_1 \) and \( C_2 \), we need initial conditions, such as:
1. The initial displacement \( x(0) = x_0 \)
2. The initial velocity \( \frac{dx}{dt}(0) = v_0 \)
Applying the initial conditions to the general solution:
1. At \( t = 0 \):
\[
x(0) = C_1 \cos(0) + C_2 \sin(0) = C_1 = x_0
\]
So, \( C_1 = x_0 \).
2. The velocity is given by the derivative of \( x(t) \):
\[
\frac{dx}{dt} = -C_1 \omega \sin(\omega t) + C_2 \omega \cos(\omega t)
\]
Evaluating at \( t = 0 \):
\[
\frac{dx}{dt}(0) = -C_1 \omega \sin(0) + C_2 \omega \cos(0) = C_2 \omega = v_0
\]
Thus, \( C_2 = \frac{v_0}{\omega} \).
### Final Solution
Putting it all together, the solution to the differential equation is:
\[
x(t) = x_0 \cos(\omega t) + \frac{v_0}{\omega} \sin(\omega t)
\]
This represents the motion of a simple harmonic oscillator with initial displacement \( x_0 \) and initial velocity \( v_0 \).