Solving a Separable 2nd Order Differential Equation

In summary, the conversation involves an equation that cannot be solved analytically, but can be solved numerically. The equation is a second order differential equation and the conversation includes a suggestion to use a clever substitution to potentially find a solution.
  • #1
Atomos
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In an investigation of a physics problem, I ran into the following equation:

d^2(y)/(dt)^2 = k * y * (y^2 + c)^-1.5

I know how to solve separable first order differential equations but this one seems to be beyond me. Assistance?
 
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  • #2
hmm I don't think that one can be solved analytically, can you settle for a numeric answer?
 
  • #3
Well, one thing you can do is multiply by y prime

[tex]y^{\prime} y^{\prime \prime} = \frac{k y y^{\prime}}{(y^2 + c)^\frac{3}{2}} [/tex]

and then integrate to get

[tex] \frac{1}{2} y^{\prime 2} = - \frac{k}{\sqrt{y^2 + c}} + A [/tex]

where A is a constant of integration.

You can then square root the y prime square, pull over all the y stuff on one side (and integrate again) to get x as some horrendous integral in y.

i.e.,

[tex]x = \int{\frac{dy}{\sqrt{2(A- \frac{k}{\sqrt{y^2 + c}})}}} [/tex]

or rather

[tex]x = \frac{1}{\sqrt{2}} \int{\sqrt{\frac{\sqrt{y^2 +c}}{A \sqrt{y^2 +c} - k }} dy} [/tex]
Other than that, I dunno.
 
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  • #4
That looks intractable. I expected there to be a "clean" or closed (or whatever you call it) solution. This equation arose from me trying to plot the position of a point mass in a field generated by another point mass. The y is the vertical position (the reference point mass is at the origin and is stationary).
 
  • #5
With the use of a clever substitution, it may yet be soluble. You never know.
 

1. What is a separable 2nd order differential equation?

A separable 2nd order differential equation is a type of differential equation that can be written in the form of y'' = f(x)g(y), where y' represents the first derivative of y with respect to x and y'' represents the second derivative of y with respect to x. This form allows the equation to be separated into two functions, f(x) and g(y), making it easier to solve.

2. How do I solve a separable 2nd order differential equation?

To solve a separable 2nd order differential equation, you can follow these steps:

  1. Separate the equation into two functions, f(x) and g(y).
  2. Integrate both sides of the equation with respect to x and y, respectively.
  3. Combine the two integrals and solve for y to find the general solution.
  4. Apply initial conditions to find the particular solution.

3. What are the initial conditions in a separable 2nd order differential equation?

The initial conditions refer to the values of y and its first derivative, y', at a specific point. These conditions are necessary to find the particular solution of the equation. They are usually given in the problem or can be determined from the context.

4. Can a separable 2nd order differential equation have more than one solution?

Yes, a separable 2nd order differential equation can have an infinite number of solutions. This is because the equation involves integration, which introduces a constant of integration. Different values of this constant can result in different solutions to the equation.

5. What are some real-life applications of separable 2nd order differential equations?

Separable 2nd order differential equations can be used to model various physical phenomena, such as the motion of a pendulum, the growth of a population, or the decay of radioactive substances. They are also commonly used in engineering, physics, and economics to describe systems that involve acceleration, velocity, and position.

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