How to solve 2nd order diff. equation for simple harmonic motion

[cjurban]

In my physics class we're talking about LC and LRC circuits, and the equations are analogous to those for SHM. However, I don't see how x=Acos(ωt+[itex]\varphi[/itex]) satisfies m(d^2x/dt^2)+(k/m)x=0. I've never done differential equations and in the book it seemed like the author just guessed and checked until he found the right solution, and this doesn't seem like a satisfactory answer. How did he solve this equation?

 

[SteamKing]

Take the assumed solution, do the requisite manipulation, and see if it satisfies the ODE. Any unknown constants can be determined using initial conditions. You have taken derivatives of trig functions, haven't you?

 

[vanhees71]

Well, solving differential equations is an art. You just have to practice it to find solutions. There is of course a lot of work done by the mathematicians to provide general theorems about their structure and that of their solutions.

In your case of circuit theory or the harmonic oscillator you have a particularly nice class of differential equations, namely a linear differential equation. Here, it's even the kind with constant coefficients. For the undamped harmonic oscillator it reads
[tex]\ddot{x}+\omega^2 x=0.[/tex]

Here the standard ansatz to find a solution is
[tex]x(t)=A \exp(\lambda t).[/tex]
Plugging this into the equation you find
[tex]\lambda^2+\omega^2=0.[/tex]
There are two solutions for [itex]\lambda[/itex], namely
[tex]\lambda_{1/2}=\pm \mathrm{i} \omega.[/tex]
Now the mathematicians have proven that any solution is given as the superposition of two linearly independent solutions, and these we just have found! So the general solution is
[tex]x(t)=A_1 \exp(\mathrm{i} \omega t) + A_2 \exp(-\mathrm{i} \omega t).[/tex]
The constants [itex]A_1[/itex] and [itex]A_2[/itex] are determined by giving initial values, e.g., position and velocity at [itex]t=0[/itex].

 

[timthereaper]

I have a book that does something similar and I'm not sure the book was guessing and checking. I think they were showing why other functions don't satisfy the given ODE.

 

[PJC01]

I can understand your confusion and frustration with the solution to the second order differential equation for simple harmonic motion. It may seem like the author simply guessed and checked until they found the right solution, but in reality, there is a systematic method for solving these types of equations.

The equation you mentioned, m(d^2x/dt^2)+(k/m)x=0, is a second order linear homogeneous differential equation. This type of equation can be solved using the technique of separation of variables. This involves separating the variables on each side of the equation and then integrating both sides to find the general solution.

In the case of simple harmonic motion, the general solution can be expressed as x=Acos(ωt+\varphi), where A is the amplitude, ω is the angular frequency, and φ is the phase angle. This solution is derived by substituting x=Acos(ωt+\varphi) into the differential equation and solving for the constants A, ω, and φ.

It is important to note that the solution x=Acos(ωt+\varphi) is not guessed, but rather derived from the mathematical principles of differential equations. This solution satisfies the differential equation because when you take the second derivative of x with respect to time, you end up with -ω^2Acos(ωt+\varphi), which is equal to -(k/m)x, as required by the original equation.

In conclusion, the solution to the second order differential equation for simple harmonic motion is not a mere guess, but rather a systematic approach based on mathematical principles. I encourage you to further explore the techniques of solving differential equations in order to fully understand the solution to this equation and its applications in physics.