Differential Equations Graphic Organizer

Differential Equations Graphic Organizer

1. Maya Faour

   • Ms. Van Dyken
     • AP Calculus AB B
       • 03/02/2025

2. Differential Equation

   • Description: Differential equations are equations that contain derivatives within them. They can contain one or multiple derivatives, the standard first-order derivative. However, they can also be second, third-, fourth-, and more ordered derivatives.
   • Example: We see differential equations all the time in this unit, whether it is to solve a differential equation to find the general or exact solution, to verify a differential equation, or to use them for real-world applications like determining the half-life of an isotope. So, a straightforward example of a differential equation is dy/dx=x^2+5x+2, and you are asked to find the general solution. You would need to find the antiderivative, in other words, of the given differential equation. It is clear that it is a differential equation because it does not say y = or f(x) =; rather, it uses the derivative notation, dy/dx =. In this case, the antiderivative would be f(x)=x³/3+5x²/2+2x+C.

3. Initial Value Problem

   • Description: An initial value problem is a lot like a general solution. The main difference is that an initial value, often a coordinate pair, is provided to solve for the constant C.
   • Example: So, for instance, looking at the same problem as was used as an example for the general solution, f(x)=x³/3+5x²/2+2x+C, the initial value problem “version” would provide an extra piece of information to determine the value of the constant C. For instance, the initial value problem could have included the f(1) = 3. Then, we could substitute x for 1 and f(x) for 3 and have C as the only variable, allowing us to solve for it. In that case, C would be -2, so the full equation would be f(x)=x³/3+5x²/2+2x-2.

4. General Solution

   • Description: A general solution is a solution to a differential function looking at a specific family of functions. In a general solution, there will be a constant C whose value is unknown without further information.
   • Example: An example is the solution determined in the example of “Differential Equation”: f(x)=x³/3+5x²/2+2x+C. It falls under the specific family of functions, the polynomials, and there is a constant C present whose value remains unknown. However, there are numerous families of functions that could be the general solution: it could be a trigonometric function, a rational function, or a logarithmic function, just as long as it represents the solution to a differential function and contains an unknown C value.

5. Exact Differential Equation

   • Description: An exact differential equation already has the derivative solved, so generally, the antiderivative can be easily found without special techniques like separation or substitution.
   • Example: Any differential equation with an isolated derivative would be considered an exact differential equation. For instance, dy/ⅆx=x^2+5x+2 would be an exact differential equation – we did not need to do anything before finding the antiderivative. Another example is dy/ⅆx=sec^2 x+csc²x – if you know the basic trigonometric derivatives, this is easy to solve and find the antiderivative. Nothing had to be done before you could find the antiderivative either, i.e., you did not need to separate the values, use substitution, etc.

6. Slope Field

   • Description: A slope field can display the slope at each point of a differential function. It does this by graphing a line at each coordinate point that represents the specific slope, with positive slopes represented by positive slopes of varying slopes, negative slopes represented by negative slopes of varying slopes, slopes of 0 represented as horizontal lines, and unknown slopes represented as vertical lines.
   • Example: Attached is shown the slope field of dy/ⅆx=x – it is easy to tell as each line has a slope of its corresponding x-coordinate. All the points with the x-coordinate of 1 also have a slope of 1, each point with the x-coordinate of 6 has a slope of 6, and all the points with the x-coordinate of -2 have a slope of -2.

7. Indefinite Integral

   • Description: An indefinite integral is a notation used to express a derivative's antiderivative finding. It is essentially like a sign that you attach to an equation to clarify that the operation to be performed on that integral is the finding of the antiderivative.
   • Example: ∫2x is an example of an indefinite integral, albeit with a relatively obvious answer of x². ∫▒〖-cos⁡x 〗 is another example of an indefinite integral, after finding the antiderivative and solving equals – sin x. Or you could have something as complex as ∫(e^2 2x^3)/cos^2⁡x . The point is that all of these are examples of indefinite integrals, and to “solve” them, you need to find their antiderivative.

8. Substitution Methods for Integration

   • Description: Integrating via substitution is a technique that can be used to find the antiderivative by performing the opposite operation of the Chain Rule. This especially helps if you have a function within another function, resulting in greater difficulty in finding the antiderivative.
   • Example: For instance, if I have the differential function ∫2x sin⁡〖x^2 〗 dx, it might be tricky figuring out how to find its antiderivative. However, the Chain Rule was used to find the derivative in the first place. It is clear since there is a function within a function, x², within the sin function. So, we can substitute a specific value with a term, often u. So, let us say I want to substitute x² with u. You also need to find du, which would be 2x dx, which we can manipulate to get dx = du/2x. Then you would substitute: ∫2x sin⁡u du/2x=∫sin⁡u du. Now it looks clearer that we know the antiderivative of: - cos u. Then we resubstitute the initial values and are done: - cos x².

9. Seperable Differential Equation

   • Description: A separable differential equation is a differential equation where f can be described as a function of x and y. So, in other words, it is a lot like an implicit equation. However, in the context of a differential equation, we want to find its antiderivative.
   • Example: So, the question is, how do you find the antiderivative of such a differential equation? The answer lies in separation. The x terms are separated to the right side of the equation, while the y terms are separated to the left side of the equation. For instance, the differential equation dy/ⅆx=x/y is a separable differential equation. To solve it, we would separate the terms: y dy=x dx. Then the antiderivative of each side is found: y² = x² + C. Then the equation is simplified, and then y is isolated: y=√(x^2+C).

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