Solving a Linear Differential Equation

Click For Summary

Discussion Overview

The discussion revolves around the classification and solution of a linear differential equation, specifically examining the equation (x + 4y^2) dy + 2y dx = 0. Participants explore its form, linearity, and potential methods for solving it, including transformations to different variables.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant attempts to express the given equation in the general form of a linear differential equation but struggles with isolating x to identify P(x).
  • Another participant challenges the classification of the equation as a differential equation, suggesting it does not meet the criteria.
  • A subsequent post clarifies that the original equation was missing an equality to zero, which is necessary for proper classification.
  • Another participant suggests a non-standard approach, proposing that the equation can be transformed into an ordinary differential equation (ODE) through a different perspective.
  • Some participants assert that the equation is not linear, with one providing a transformation that expresses the equation in a linear form for x as a function of y.

Areas of Agreement / Disagreement

Participants express disagreement regarding the classification of the equation as linear or as a valid differential equation. Multiple competing views are presented, and the discussion remains unresolved.

Contextual Notes

There are limitations in the assumptions made about the equation's form, and the discussion highlights the dependency on definitions of linearity and differential equations. The transformation steps and their implications are not fully resolved.

misogynisticfeminist
Messages
370
Reaction score
0
I've got a linear DE here,

[tex](x + 4y^2) dy + 2y dx =0[/tex]

I've tried to put it in the general form of a linear equation, and I would get,

[tex]\frac {dy}{dx} + \frac {2y}{x+4y^2} = 0[/tex]

but I have problems isolating the x, so that I would get the [tex]P(x)[/tex] in the general form.
 
Last edited:
Physics news on Phys.org
This doesn't make sense. If "DE" stands for "differential equation" then no, you don't have a DE here.
 
^ sorry, its supposed to be equals to 0, i forgot to add that...
 
Think outside the box for a moment. There's a very easy way to turn this into an ODE.
 
Your D.E. is not linear, either.
 
The way you have written it
[tex]\frac {dy}{dx} + \frac {2y}{x+4y^2} = 0[/tex]
it is not linear.
However, you can write
[tex](x + 4y^2) dy + 2y dx =0[/tex]

as
[tex]\frac{dx}{dy}= -\frac{x+4y^2}{2y}= -\frac{1}{2y}x- 2y[/tex]
which is a linear d.e. for x as a function of y.
 

Similar threads

Undergrad How do I solve this first order second degree differential equation?
  • · Replies 7 ·
Replies
7
Views
4K
Undergrad Why Lagrange’s method of solving Pp + Qq=R works?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Explanation of all "its linearly independent derivatives"
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Expressing a differential equation into a different format
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Integral-form change of variable in differential equation
  • · Replies 1 ·
Replies
1
Views
2K
MHB Can You Solve This System of Differential Equations with Initial Values?
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Proving a basis is a basis for homogeneous linear differential equation with constant (complex) coefficients
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Direction Fields and Isoclines
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Second order non-homogeneous linear ordinary differential equation
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Maxwell's equations PDE interdependence and solutions
  • · Replies 5 ·
Replies
5
Views
3K