First Order Linear Differential Equations

In summary, the conversation discusses a cosmological model in which the Friedmann equation is represented by L^2 (a')^2 = a^2 - 2a^2 + 1, with L as a positive constant, a dot representing time differentiation, and the initial condition of a(0) = 1. The conversation also addresses finding the units of L and proving that the universe described by this model is never smaller than a certain minimum size. The method of solving the equation is also discussed, with different opinions on the approach. However, there is uncertainty and confusion about the validity and relevance of the equation.
  • #1
mit_hacker
92
0

Homework Statement



In a particular cosmological model,
the Friedmann equation takes the form L^2 (a')2 = a^2 − 2a^2 + 1, where L is a positive constant,
the dot denotes time differentiation, and the initial condition is a(0) = 1. What are the units of
L? Show, without solving this equation, that the universe described by this model is never smaller
than a certain minimum size. Now solve the equation and describe the history of this universe.

Homework Equations





The Attempt at a Solution



I basically considered this as an autonomous equation and found the critical points. Once A takes those values, the derivative will be 0 so the value of the function will not change. In class however, my tutor discussed some other weird (in my opinion) method of solving the problem which simply went over my head. Can someone please help me confirm whether I'm correct?

Also, I don't see how we can "describe the history of this universe" by solving this equation. Please advise!

Thank-you very much for your kind co-operation!
 
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  • #2
If I put a(0)=1 into your equation it looks like a'(0)=0. So the solution is the trivial solution a(t)=1 for all t. It hardly matters how you solve something like that. I suspect there is either a typo or unclear notation. Can you clarify what the ODE actually is??
 
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  • #3
Did you mean to write (a')^2? Also what is the point of writing a^2 − 2a^2 instead of −a^2?
 
  • #4
Of course, the whole question makes no sense without saying what the variable a reoresents physically! The "diameter" of the universe?

Also I notice this is titled "First Order Linear Differential Equations". While that differential equation is first order, it definitely is not linear!
 
  • #5
Ouch!

I agree with all of you but that's lecturer what it is. It's probably one of those questions which the lecturer set by mistake or just for the sake of it.

Evidently, everything about it is wrong so I guess I'll just ignore the question. Thanks you guys for your help!
 

FAQ: First Order Linear Differential Equations

What is a first-order linear differential equation?

A first-order linear differential equation is an equation that relates a function and its derivative in a linear form. It can be written in the form of y' + p(x)y = q(x), where y is the dependent variable, x is the independent variable, p(x) and q(x) are functions of x, and y' denotes the derivative of y with respect to x.

What is the general solution to a first-order linear differential equation?

The general solution to a first-order linear differential equation is the most general form of the solution that satisfies the given equation. It is usually expressed in terms of an arbitrary constant, C, which can take on any value. The general solution can be found by integrating both sides of the equation and then solving for y.

What is the difference between a linear and a nonlinear differential equation?

A linear differential equation is one in which the dependent variable and its derivatives appear in a linear form, as in y' + p(x)y = q(x). In contrast, a nonlinear differential equation is one in which the dependent variable and its derivatives appear in a nonlinear form, such as y' + p(x)y^2 = q(x). The solution to a linear differential equation is usually easier to find than that of a nonlinear one.

How are first-order linear differential equations used in real-world applications?

First-order linear differential equations are used in various fields of science and engineering to model and predict the behavior of systems that involve change over time. They are particularly useful in the fields of physics, chemistry, biology, and economics, to name a few. Examples of real-world applications include population growth, chemical reactions, and electrical circuits.

What are the steps for solving a first-order linear differential equation?

There are several methods for solving a first-order linear differential equation, but the most common approach involves the following steps:

  1. Identify the dependent variable, independent variable, and functions p(x) and q(x) in the given equation.
  2. Separate the variables by moving all terms involving the dependent variable to one side and all other terms to the other side.
  3. Integrate both sides of the equation with respect to the independent variable.
  4. Solve for the dependent variable, including the arbitrary constant C in the solution.
  5. If initial conditions are given, use them to find the specific solution that satisfies the given conditions.

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