Solving First Order Linear ODE: dy/dx = y/x + tan(y/x)

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Homework Help Overview

The discussion revolves around solving the first-order linear ordinary differential equation (ODE) given by dy/dx = y/x + tan(y/x). Participants explore various methods and substitutions to approach the problem.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to identify the nature of the ODE and expresses uncertainty about applicable methods, questioning whether a substitution might be useful. Some participants suggest the substitution u = y/x as a potential approach, leading to a reformulation of the equation.

Discussion Status

Participants have engaged in a productive exchange, with one confirming that the suggested substitution leads to a separable form of the equation. There is an acknowledgment of the effectiveness of the substitution, despite initial hesitations about its applicability.

Contextual Notes

There is a discussion about the classification of the ODE, with participants noting that it does not fit typical forms such as homogeneous equations or linear equations in standard form. The exploration of substitutions indicates a search for a viable method to solve the equation.

whatisreality
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Homework Statement


Solve dy/dx = y/x + tan(y/x)

Homework Equations

The Attempt at a Solution


Not separable, as far as I can tell. It's not homogeneous, since for the tan term f(λx,λy) = tan(λy/λx) = tan(y/x) ≠ λtan(y/x). It's also not of the form dy/dx + P(x)y = Q(x), because I don't think Q(x) should involve y. And that completes the list of methods I know, none of which I can use! How do you solve this?! Is there a substitution I should make?
 
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Even though this is not homogeneous, seeing that "y/x" my first thought would be to try the substitution u= y/x.

Then y= xu so that dy/dx= u+ x du/dx. The differential equation becomes u+ x du/dx= u+ tan(u) so that x du/dx= tan(u).

That is separable.
 
You could try making a change in the dependent variable, from y to u, where u= y/x
 
OK, that substitution works! I thought it was only for homogeneous equations. Thanks :)
 

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