Solving separable 2nd order DE

In summary, the book's method for solving DEs of the form \frac{d^2 y}{dx^2} = f(y) is to assume the following two equations hold: $$\frac{d^2 y}{dx^2} = f(y)$$ $$\frac{dy}{dx} = f(y)$$
  • #1
phantomvommand
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Homework Statement
please see attached pictures.
Relevant Equations
Integration, Substitution
Screenshot 2021-07-25 at 5.29.46 PM.png

This is a physics problem from Griffith's Electrodynamics. I'm mainly asking about the math here. I found the DE in the box at part (d).
To solve it, I did:
##\sqrt V {d^2 V} = \beta dx^2##
Integrating twice:
##\frac {4} {15} V^{2.5} = \beta x^2/2##

Why is my method wrong?

Thanks for the help.
 
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  • #2
Write your second derivative as [itex]\frac{d}{dx} \left( \frac{dV}{dx} \right) [/itex] and it should be clear that when you try to separate it using your method, what you actually get is:

[tex]\sqrt{V} \, d (\frac{dV}{dx}) = \beta \, dx [/tex]
 
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  • #3
You just cannot integrate this equation is $$\sqrt{V}d^2V=\beta dx^2$$ because , me at least , doesn't know how an integral of the form $$\int f(V)d^2V$$ is defined or an integral of the form $$\int f(x)dx^2$$ is defined.
That is because you have ##d^2V## there instead of ##dV## and ##dx^2## instead of ##dx##.
$$\int f(V)dV$$ and $$\int f(x) dx$$ are well defined, but not the integrals at the starting of my post, those are just not well defined.

What you do is that you assume not only that the integrals are well defined but also that the following equality holds $$\int f(V)d^2V=\ (\int f(V) dV \ )dV$$ (and something similar for the $$\int f(x)dx^2=\ (\int f(x) dx \ )dx$$.

I challenge you to prove the above equalities, there is no way to prove them, simply because the left hand side is not well defined
 
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  • #4
BTW, the method the book uses is the standard way to solve DEs of the form [itex]\frac{d^2 y}{dx^2} = f(y)[/itex]

Define the first derivative as a new variable, eg [itex] v = \frac{dy}{dx}[/itex].
Now in terms of [itex]v[/itex] the original DE can be written as:
[tex] \frac{dv}{dx} = f(y)[/tex]
Applying the chain rule you get:
[tex]\frac{dv}{dy} \, \frac{dy}{dx} = f(y)[/tex]
[tex] v \frac{dv}{dy} = f(y)[/tex]
Which separates nicely to give:
[tex] v \, dv = f(y) \, dy[/tex]
 
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FAQ: Solving separable 2nd order DE

What is a separable 2nd order differential equation?

A separable 2nd order differential equation is a type of differential equation where the dependent variable and its derivatives can be separated into two functions, each containing only one variable. This allows for the equation to be solved by integrating both sides separately.

How do you solve a separable 2nd order differential equation?

To solve a separable 2nd order differential equation, you first need to separate the variables on both sides of the equation. Then, integrate both sides separately. This will result in a general solution. To find the particular solution, you will need to use initial conditions or boundary conditions.

What are the steps to solve a separable 2nd order differential equation?

The steps to solve a separable 2nd order differential equation are as follows:

  1. Separate the variables on both sides of the equation
  2. Integrate both sides separately
  3. Apply initial conditions or boundary conditions to find the particular solution

Can a separable 2nd order differential equation have multiple solutions?

Yes, a separable 2nd order differential equation can have multiple solutions. This is because the general solution obtained after integrating both sides separately will have a constant of integration, which can take on different values for different solutions. To find the unique solution, you will need to use initial conditions or boundary conditions.

What are some real-world applications of solving separable 2nd order differential equations?

Solving separable 2nd order differential equations is used in various fields of science and engineering, such as physics, chemistry, and biology. Some real-world applications include modeling the motion of a pendulum, predicting population growth, and analyzing the decay of radioactive substances.

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