Second order differential equation

In summary, the conversation discusses how to turn a second order homogeneous equation into an expression of displacement relevant to time. The process involves using the chain rule and high school calculus to solve the equation. However, it is noted that the first method is simpler and more standard.
  • #1
hallic
6
0
I've created a second order homogeneous equation from my orginal data
m(d^2x/dt^2) + kx = 0
how can I turn it into a expression of displacement relevant to time?
 
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  • #2
[tex]\frac{d^2x}{dt^2} = \frac{-kx}{m}[/tex]

Now,

[tex]\frac{d^2x}{dt^2} = \frac{dv}{dt} = \frac{dv}{dx}\frac{dx}{dt} = v\frac{dv}{dx} = \frac{d(\frac{1}{2}v^2)}{dv}\frac{dv}{dx} = \frac{d(\frac{1}{2}v^2)}{dx}=\frac{d}{dx}\left(\frac{1}{2}\left(\frac{dx}{dt}\right)^2\right)[/tex]

By applying the chain rule twice.

So,

[tex]\frac{d}{dx}\left(\frac{1}{2}\left(\frac{dx}{dt}\right)^2\right) = \frac{-kx}{m}[/tex]

Hopefully you can now solve this as a differential equation
 
  • #3
Isn't it simpler to solve the first one, it being standard? [tex]x=ACos(\sqrt{\frac{k}{m}}t+\phi_{0})[/tex]
 
  • #4
Haha yeah, I guess so. But since I haven't learned the theory for second order differential equations, this is a way using high school calculus.

If you go through it, it comes out to be the same answer you have
 

Related to Second order differential equation

What is a second order differential equation?

A second order differential equation is a mathematical equation that involves the second derivative of a function. It is commonly used in physics and engineering to describe the motion of objects.

What is the difference between a first and second order differential equation?

The main difference between a first and second order differential equation is the number of derivatives involved. A first order differential equation involves the first derivative of a function, while a second order differential equation involves the second derivative.

What are some real-world applications of second order differential equations?

Second order differential equations have many real-world applications, including in physics (such as describing the motion of a pendulum or a spring), engineering (such as modeling electrical circuits), and economics (such as predicting the growth of a population).

How do you solve a second order differential equation?

Solving a second order differential equation involves finding a function that satisfies the equation. This can be done using various techniques, such as separation of variables, substitution, or solving for an integrating factor.

What is the order of a differential equation?

The order of a differential equation is the highest derivative present in the equation. For example, a second order differential equation has a second derivative, while a third order differential equation has a third derivative.

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