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In summary, the conversation discussed topics and methods learned in a differential equations course, including separable equations, direction fields, exact equations, and linear and non-linear equations. The quickest way to tell if an equation is linear is to check if it follows the standard form of y' + p(x)y = r(x), where p(x) and r(x) are functions of x only or constants. A homogeneous equation has the form y' = f(x,y) where the right side can be expressed as a function of the ratio y/x alone. A non-homogeneous equation has r(x) as a function of x only. Drawing direction fields involves using isoclines and tangents to determine the direction of solution curves. Additional resources,
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Hi,

I am taking a course in differential equations.

So far we have learned the following topics/methods:

- Separable equations
- Direction fields
- Exact equations
- Linear ODE's - Homogeneous & Non-Homogeneous equations

1. How do you know if an equation is linear or non-linear. What is the quickest way to tell?
>> In order to be linear, does it have to have the form: y' + p(x)y = r(x) (Known as the Standard Form). Where p(x) and r(x) are functions of x only or constants?

2. When y' + p(x)y = 0; where r(x) = 0. Is this a Homogeneous equation? Why?

3. When y' + p(x)y = r(x); where r(x) is a function of x only or a constant. Is this a Non-Homogeneous equation? Why?

4. Finally, can someone explain a methodical way to draw direction fields for a given ODE. It has something to do with isoclines. When you draw the direction fields you get tangents (slope lines) which define the direction of the solution curves y(x). Can someone give me an idea how to do this with an example.

Thanks.

1. More or less...it has to have the format:
A(x)d^n +An-1(x)d^n-1...=F(x)
They'll probably only make you know this enough to put an equation into Standard Form that may not initially "appear" to be in Standard Form.

2. A homogeneous equation will be in the form of y'=f(x,y) where the right side can be expressed as a function of the ratio y/x alone...one way that usually works as a check is to see if the sum of the powers are the same for each term.
So: x^3 + x^2y = f(x,y) Both terms add up to degree 3 (3 and 2+1=3)

I've also seen its definition written like; A homogenous equation is an equation that can be rewritten into the form having zero on one side of the equal sign and a homogeneous function of all the variables on the other side.

I shy away from any official "definition" for a mathematical term...if you give an absolute for something involving math...a math guy will show you where you're wrong. ..lol3. r(x) is a function of x only.

4. There should be some examples in your book. My diff eq professor spent about 5 minutes with direction fields...said they were nice to know...and then told us he wouldn't be testing us on them. So, fearful that I may learn something I don't need to know, I never paid attention. lol

I believe MIT's opencourseware for differential equations with Arthur Mattuck shows field lines in some detail the first lecture. You can find them on youtube.

1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is used to model many natural phenomena in various fields such as physics, engineering, and economics.

2. What are the types of differential equations?

The types of differential equations include ordinary differential equations (ODEs), which involve a single independent variable, and partial differential equations (PDEs), which involve multiple independent variables. Other types include linear and nonlinear differential equations, and first-order and higher-order differential equations.

3. How do I solve a differential equation?

The method for solving a differential equation depends on its type and complexity. Some common techniques include separation of variables, substitution, and using integrating factors. Numerical methods, such as Euler's method and Runge-Kutta methods, can also be used to approximate solutions.

4. Why are differential equations important?

Differential equations are important because they provide a way to mathematically model and analyze real-world problems. They are used in many fields to make predictions, understand patterns and behaviors, and design solutions to complex systems.

5. Can differential equations be solved analytically?

Some differential equations can be solved analytically, meaning an exact, closed-form solution can be found. However, for many equations, an analytical solution is not possible and numerical methods must be used to approximate the solution. Additionally, some equations may have no solution or multiple solutions.

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