Solving Complex Differential Equations: What Are the Steps?

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Homework Help Overview

The discussion revolves around solving complex differential equations, specifically focusing on equations that involve trigonometric functions and separable forms. Participants are exploring various methods and approaches to tackle these equations.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the separability of certain equations and share identities that may assist in solving them. There are attempts to identify the nature of specific differential equations, such as questioning if one resembles a Riccati equation. Suggestions include manipulating equations by squaring or differentiating to facilitate separation of variables.

Discussion Status

The discussion is active, with participants providing insights and suggestions for approaching the problems. Some have expressed uncertainty about their methods, while others have indicated progress in solving specific equations. There is a mix of interpretations regarding the best strategies to employ.

Contextual Notes

Participants are working under the constraints of homework assignments, which may limit the information they can share or the methods they can use. There is a recognition of the complexity of the equations being discussed, which may contribute to the varied approaches and interpretations.

Icebreaker
How do I go about solving

x^2y' - cos (2y) = 1

This is unlike anything I've seen so far. Or so I think.
 
Last edited by a moderator:
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Icebreaker said:
How do I go about solving

x^2y' + cos (2y) = 1

This is unlike anything I've seen so far. Or so I think.

It's separable right?
 
saltydog said:
It's separable right?
Yes
Trig is your friend.
as a matter of personal taste I enjoy the use of the identity
1-cos(2y)=2(sin(y/2))^2
 
Ok, I got it, thanks for the help.

I got y = arctan ( c = 2/x)
 
Last edited by a moderator:
18 down, 2 more to go:

y' = (4x + 2y - 1)^(1/2)

and

(1-xy)y' + y ^2 + 3xy^3 = 0

The last one kinda looks like Riccati, but I can't get it to work.

edit: I got the second one. One more ODE remains... I don't know how to approach that one.
 
Last edited by a moderator:
Icebreaker said:
18 down, 2 more to go:

y' = (4x + 2y - 1)^(1/2)

and

(1-xy)y' + y ^2 + 3xy^3 = 0

The last one kinda looks like Riccati, but I can't get it to work.

edit: I got the second one. One more ODE remains... I don't know how to approach that one.

Regarding:

[tex]\frac{dy}{dx}=\sqrt{4x+2y-1}[/tex]

How about squaring it for starters. Then let:

[tex]p=\frac{dy}{dx}[/tex]

Differentiate it to get an ODE in p. Then separate variables.
 

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