First order differential equation

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SUMMARY

The discussion centers on solving the first-order differential equation \(\frac{dy}{dx}\sin x + y \sec x = \cos^2 x\). The user expresses uncertainty about finding an integrating factor or differentiating to simplify the equation. A key insight provided is to consider the equation in the form \((u y)' = u' y + u y'\), leading to the solution \(y = \frac{v}{u}\), where the choice of functions \(u\) and \(v\) is crucial. Additionally, the hint regarding the derivative of \(\tan(x)\) suggests a potential direction for solving the equation.

PREREQUISITES
  • Understanding of first-order differential equations
  • Familiarity with trigonometric functions and their derivatives
  • Knowledge of integrating factors in differential equations
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the method of integrating factors for first-order differential equations
  • Learn about the application of trigonometric identities in solving differential equations
  • Explore the concept of separable equations and their solutions
  • Investigate the relationship between derivatives and trigonometric functions, specifically \(\tan'(x)\)
USEFUL FOR

Students studying calculus, particularly those focusing on differential equations, as well as educators seeking to clarify concepts related to first-order differential equations and their solutions.

Xenith
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I am asked to find the general solution to:

[itex]\dfrac{dy}{dx}\sin x + y \sec x = \cos^2 x[/itex]

I don't quite know where I am going with this one; by simply looking at it, I can't seem to see what I would differentiate in order to get the left side and equally, I don't know if dividing through by and finding the integrating factor is a good idea either.

I need a nudge in the correct direction really! Sorry if I am not spotting anything glaringly obvious. I have just started looking at this topic a few hours ago.

Thanks very much in advance
 
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By the way, I am still studying my A levels so this isn't really advanced at all ;)
 
You should think of this as being
(u y)'=u' y+u y'=v'
easily solved as
y=v/u
choosing u and v are what matter.

hint: What is
tan'(x)
 

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