First Order, Non-Linear DE - Not Seperable

CIn summary, the given equation involves trigonometric functions and variables that cannot be separated, making it difficult to solve. The modified equation may be solved by finding its derivative with respect to x and y and using the usual methods of solving exact equations.
  • #1
Berkshire
3
0
Hello, I'm studying for a test and this is a question on a practice test...

cos(x)+y^2+(2yx-1)y'=0


I can't separate the variables (it's not homogeneous, either), this isn't exact and bernoulli won't work...

dy/dx=-cos(x)/(2yx-1)-y^2/(2yx-1)

I changed the equation so it would look like this but I can't simplify it any more than that and I can't just take the integral of it here...If anyone could give me some help with this problem it would be much appreciated. Thanks!
 
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  • #2
Looks like an exact equation, if it's on your test you should know how to solve it. That said, it doesn't look like the solution is anything obvious so you'll have to go through the usual P = df/dx and Q = df/dy
 
  • #3
Oh you're right, thanks!
 
  • #4
Hi !

cos(x)+y²+2yxy'-y'=0
cos(x)+(x y² -y)'=0
 
  • #5


It seems like you have already done a good job in trying to manipulate the equation to make it easier to solve. However, as you have mentioned, this first order non-linear differential equation is not separable and cannot be solved using the exact or Bernoulli method. This means that you will need to use a different approach to solve it, such as using an integrating factor or a substitution method. I would suggest consulting with your textbook or instructor for specific techniques and strategies to solve this type of equation. Good luck on your test!
 

1. What is a first order, non-linear differential equation?

A first order, non-linear differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is called "first order" because it involves only the first derivative of the function, and "non-linear" because the function and its derivatives are not directly proportional to each other.

2. How is a first order, non-linear differential equation different from a linear differential equation?

In a linear differential equation, the function and its derivatives are directly proportional to each other. This means that the equation can be written in the form of y=mx+b, where m is the slope and b is the y-intercept. In a non-linear differential equation, the relationship between the function and its derivatives is not directly proportional.

3. What are some examples of first order, non-linear differential equations?

Some examples of first order, non-linear differential equations include the logistic equation, the Lotka-Volterra equation, and the predator-prey equation. These equations are commonly used in population dynamics and modeling of biological systems.

4. How do you solve a first order, non-linear differential equation?

Solving a first order, non-linear differential equation involves finding an expression for the function that satisfies the given equation. This can be done using various methods such as separation of variables, substitution, or using an integrating factor.

5. What are the applications of first order, non-linear differential equations?

First order, non-linear differential equations have a wide range of applications in various fields such as physics, biology, economics, and engineering. They are used to model complex systems and predict their behavior over time. Some common applications include population growth, chemical reactions, and circuit analysis.

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