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I've just started learning Differential Equations and am having general difficulties with a bit of concepts and on how to actually get started. The problem I have is that the books and sources I find always launch into slope fields, and while I get what a field is and how it works. I don't understand how it relates to a differential equation, and more importantly how to solve them. We started with very basic examples and I am already at a loss of what to do.

Essentially my questions boil down to something like this:

  1. What does a slope field represent? i.e. What are the lines indicating or plotting?

  2. How do they relate to differential equations?

  3. What are the general steps to solving a very basic differential equation? Such as: $\frac{dy}{dx} = y + 5 $ ($\frac{dy}{dx}$ = $y'$ right?)

Please keep in mind that I have no knowledge of differentials, just calculus, including mulitvariable. (i.e. Calc I-III in the states).

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(1). In a slope field, the direction of the arrow at each point represents the direction of the derivitive (or slope, hence its name), and the length sometimes represents the magnitude of the derivative at the point.

This enables you to draw an approximate solution to the differential equation starting at any point, just by following the arrows.

(2). As stated above, by knowing the slope of the derivative at each point (assuming you can compute $\frac{dy}{dy}$ for any $x$ and $y$), you can get a feel for what the solutions look like.

(3). There are many, many, many techniques. I like integrating factors.

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(1). & (2). Slope field (also called vector field or direction field) is a graphical representation of the solutions of a first-order differential equation. It is useful because it can be created without solving the differential equation analytically.

(3). There are different forms of differential equations, therefore there are diifferent ways to solve them.

One way, with taht you can solve the equation you wrote is the seperation of variables.

You get at the one side of the equality everything that relates to $y$ and at the other everything that relates to $x$.

Then you integrate.

$$\frac{dy}{dx}=y+5 \\ \Rightarrow \frac{dy}{y+5}=dx \\ \Rightarrow \int \frac{1}{y+5} dy=\int dx \\ \Rightarrow \ln{|y+5|}=x+c $$

Now, to get the solution of the differential equation in the form $y(x)= \dots$, you have to solve for $y$.

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Watch this video if you are unsure about slope fields. You are basically just plotting the arrows in laymen's terms. There will be a direction and magnitude of an arrow at a point, i.e. a vector: Check out the video below: It may be easier for you to visualize someone actually plotting the slope fields than through text: https://www.youtube.com/watch?v=eKXtN0SLNLU

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