First order differential equation

In summary, the conversation discusses finding the general solution to the differential equation dy/dx sin x + y sec x = cos^2 x. The speaker is struggling with differentiating and finding the integrating factor, and asks for guidance. The solution is suggested to be (u y)' = u' y + u y' = v', which can be easily solved by choosing appropriate u and v values. A hint is given to consider the derivative of tan x + sec^2 x.
  • #1
Xenith
2
0
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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  • #2
By the way, I am still studying my A levels so this isn't really advanced at all ;)
 
  • #3
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)
 

1. What is a first order differential equation?

A first order differential equation is a mathematical equation that involves an unknown function, its derivative, and an independent variable. It represents the rate of change of a quantity over time or space.

2. How do you solve a first order differential equation?

To solve a first order differential equation, you can use various methods such as separation of variables, substitution, or integrating factors. The specific method used depends on the form of the equation.

3. What is the initial condition in a first order differential equation?

The initial condition in a first order differential equation is the value of the unknown function at a specific point or time, usually denoted as y(0). It is used to find the particular solution of the differential equation.

4. Can a first order differential equation have multiple solutions?

Yes, a first order differential equation can have multiple solutions. This is because the general solution of a first order differential equation involves an arbitrary constant, which can take on different values to give different solutions.

5. What are some real-world applications of first order differential equations?

First order differential equations have various applications in science and engineering, such as in modeling population growth, chemical reactions, and electrical circuits. They are also used in economics, biology, and many other fields to understand and predict real-world phenomena.

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