The Harmonic Oscillator

Non homogeneous Linear differential equations

Consider the non-homogeneous second-order differential equation \(y\prime\prime + py\prime + qy = f(X)\) where \(f(x) \neq 0\)
The general solution to this differential equation is given by
\(y(x) = y_{h}(x)+y_p(x)\)
Where \(y_h(x)\) is the general solution to the complementary homogeneous equation \(y\prime\prime + py\prime +qy=0\) and \(y_p(x)\) is any particular solution. When \(f(x)\) involves simple functions, we can attempt to 'guess' \(y_p(x)\) by the method of undetermined coefficients
- If f(x) is a polynomial of degree \(n\), we let \(y_p\) be an arbitrary \(n\)th degree polynomial with undetermined coefficient
- If f(x) contains an exponential \(e^kx\), then we try \(y_{p}=Ae^{kx}\), where \(A \in \mathbb{R}\) is an undetermined coefficient
- if f(x) contains either \(cos(kx) = \Re(e^{ikx})\) or \(sin(kx) = \Im(e^{ikx})\), we can instead solve for the real and imaginary solutions, respectively, of tghe complex equation \(y\prime\prime + my\prime+n\prime=e^{ikx}\) by trying \(y_{p}=Ae^{ikx}\), where \(A \in \mathbb{C}\) is an undetermined coefficient.
If any term of the trial solution is already a solution of the complementary equation, multiply the trail solution by \(x\)
Given a solution \(y_{n}(x) = c_{1}y_{1}(x) + c_{2}y_{2}(x)\)
\(y_{p}(x) = u(x)y_{1}(x)+v(x)y_{2}(x)\)
where the functions of \(u(x)\) and \(v(x)\) are given by
\(u(x)=-\int \frac{y_{2}f}{y_{1}y_{2}\prime - y_{1}\prime y_{2}}dx\)
\(v(x)=\int \frac{y_{1}f}{y_{1}y_{2}\prime - y_{1}\prime y_{2}}dx\)

Simple Harmonic Motion

A particle is said to be in simple if it acceleration is directly proportional to its displacement
\(\ddot{x} = -\omega^{2}x\)
The constant \(\omega\) denotes the angular frequency of the oscillation.
Consider a damped oscillator in which a resisting force proportional to velocity acts against an oscillating object.

Consider a damped oscillator in which a resisting force is proportional to velocity acts against an oscillating object.
\(\ddot{x}=-\omega^{2}x- 2 \gamma \dot{x}\)
\(\ddot{x} + \omega^{2}x + 2 \gamma \dot{x} = 0\)
\(\lambda = \frac{-2\beta \pm \sqrt{4 \gamma - 4\omega^{2}}{2}}= -\gamma \pm \sqrt{\gamma^{2}-\omega^2}\)
The nature of the object's motion now depends on the value of the discriminant \(\Delta = \gamma^2-omega^2\)
The nature of the object's motion now depends on the value of the discriminant \(\Delta = \gamma^2-\omega^2\)
- If \(\omega^{2} < \gamma^{2}\), we have real roots \(\lambda_1,\lambda_{2}\in\mathbb{R}\)
- \(x(t) = c_{1}e^{\lambda_{1}t} + c_{2}e^{\lambda_{2}t}\)
- Which describes the motion of an overdamped oscillator
- If \(\omega^{2} = \gamma^{2}\), we have a repeated real root \(\lambda\in\mathbb{R}\):
- \(x(t) = c_{1}e^{\lambda t} + c_{2}te^{\lambda t}\)
- This describes a critically damped oscillator
- If \(\omega^{2} > \gamma^{2}\), we have a complex conjugate roots \(\lambda = \alpha \pm i\beta \in \mathbb{C}\)
- \(x(t)=c_{1}e^{\lambda t}cos(\beta t) + c_{2}e^{\lambda t}sin(\beta t)\)
- This describes the motion of an underdamped oscillator