Solution to a second order differential equation

In summary, the particle is released from rest at y=1 and obeys the equation of motion 4p \frac{dp}{dy} \pm 2 p^2 +y=0.
  • #1
Taylor_1989
402
14
I have currently been reading a book called 'Mathematical Methods In Physical Sciences'. Whilest I was looking at the differential section I came across a differential which I have never thought about before, which is of the form:
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Directly from book

$$m\left(\frac{d^2y}{dt^2}\right)\pm l\left(\frac{dy}{dt}\right)^2+ky=0$$where the plus or minus sign must be chosen correctly at each stage of the motion
so that the retarding force opposes the motion. Let us solve the following special
case of this problem. Discuss the motion of a particle which is released from rest at
the point y = 1 when t = 0, and obeys the equation of motion

$$4\left(\frac{d^2y}{dt^2}\right)\pm 2\left(\frac{dy}{dt}\right)^2+y=0$$
-----------------------------------------------------------------------------------------------------------------------------------------And whilst I was reading how to solve it but it sort of breaks off, it starts explain the Bernoulli method then talks about the transcendental function, as stops. The actual equation in referring damped harmonic motion on a spring except that drag force is now of the form ##F=-bv^2## I assume. So I have tried to do a bit of research and not come up with much, I did think this could be done numerically by series solutions but I am interested to how this equation is solved through transcendental methods, and was wondering if some could explain the the approach on how to solve or even refer me to a bit of literature that could help me understand. This is my first time really tackling this type of differential so it a new concept and one I would like to understand.
 
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  • #2
Is ##y## a function of ##x##? Or is it an independent variable?
 
  • #3
Dewgale said:
Is ##y## a function of ##x##? Or is it an independent variable?
In there example y is the height to which the a particle from rest is released at t=0, also sorry I write the equation in the book form to make it clearer.
 
  • #4
My approach would be the following: you can try making the substitution ##p = \frac{dy}{dt}##. Using chain rule, you can find that ##\frac{dp}{dt} = \frac{dp}{dy}\frac{dy}{dt} = p \frac{dp}{dy}##. This gives you

$$ 4p \frac{dp}{dy} \pm 2 p^2 +y = 0$$

This is now a first-order equation. Try and solve it from here. Hope this helps, and hopefully it's what you were looking for.

Divide through by ##4p##.
 
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FAQ: Solution to a 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 to model systems in physics, engineering, and other scientific fields.

What is the solution to a second order differential equation?

The solution to a second order differential equation is a function that satisfies the equation and its initial conditions. It is typically expressed in terms of the independent variable and any arbitrary constants.

What methods can be used to solve a second order differential equation?

There are several methods that can be used to solve a second order differential equation, including separation of variables, variation of parameters, and Laplace transforms. The specific method used depends on the form and complexity of the equation.

How do initial conditions affect the solution of a second order differential equation?

Initial conditions, also known as boundary conditions, are values that are specified for the dependent variable and its derivatives at a particular point. These conditions can greatly affect the solution of a second order differential equation and must be taken into account during the solving process.

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

Second order differential equations have a wide range of applications in various fields such as physics, engineering, economics, and biology. They can be used to model the motion of objects, electrical circuits, population growth, and many other phenomena.

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