A mathematical expression combining a function and one or more of its derivatives If the differential equation only involves X's, then... Steps to solving differential equations The "y" in the half-life is the? What is a differential equation?

What is a differential equation?

You must have a point. The purpose of the separable differential equation is to To find the C of a differential equation, what must you have to solve? Separable Differential Equation What is a general solution?

To find the C of a differential equation, what must you have to solve?

U-substitution - Integration by parts - Integration using trigonometric identities - Trigonometric substitution - Integration of rational functions by partial fractions Slope is determined by ________. What are the different ways of integrating you might have to use to solve a differential equation? What happens to the eᶜ or how can it be broken down further? If there are no parallel segments within a slope field, that means...

What are the different ways of integrating you might have to use to solve a differential equation?

Exponentiating Ex.) ln I y I = 2x + C e e ↓ ↓ I y I = e²ˣ ⁺ ᶜ What is a solution curve? How can you undo a natural log? (ln) Example problem How do you create a slope field?

How can you undo a natural log? (ln)

The exponential can be rewritten as a product. Ex.) e²ˣ ⁺ ᶜ ---> e²ˣ • eᶜ If the differential equation only involves Y's, then... The "t" in half-life and exponential is the? When you have exponential, how can you further break down the exponential? When you are working out a separable differential equation, you always want to change the y' to...

When you have exponential, how can you further break down the exponential?

Because C is a constant and "e" is also a number, C eats any constants that are added or multiplied to. So, eᶜ just becomes one large constant. (C) Also means that the + or - is not needed. Ex.) eᶜ ---> C "The rate of change of a quantity (y) is directly proportional to the amount present." ------> translated to calculus The "t" in half-life and exponential is the? What happens to the eᶜ or how can it be broken down further? Example problem

What happens to the eᶜ or how can it be broken down further?

Use a point given, plug it in and solve for C. How to solve exponential growth problem when it fits the general solution (y = Ceᵏᵗ) Once you have fulfilled all the steps to solve for the general solution, how do you find the particular solution? If the differential equation only involves Y's, then... Cosine Function Equation

Once you have fulfilled all the steps to solve for the general solution, how do you find the particular solution?

Chapter 5 AB AP Calc -

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Calculus - Geometry

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A mathematical expression combining a function and one or more of its derivatives

The general solution includes all possible solutions - Has the " + C " included - "family of functions"

A Particular Solution of a differential equation is a solution obtained from the General Solution by assigning specific values to the arbitrary constants. - You must find C

You must have a point.

1. Put all of the y terms from the equation in one side and all of the x terms on the other. 2. Integrate each side. - For this step you may have to use different methods of integration depending on the equation you have to integrate 3. Solve for y to obtain a general solution. 4. If an initial condition is given, apply the value to the general solution and find the value of the unknown constant C. 5. Obtain the particular solution by plugging the value of c into the general solution.

U-substitution - Integration by parts - Integration using trigonometric identities - Trigonometric substitution - Integration of rational functions by partial fractions

Exponentiating Ex.) ln I y I = 2x + C e e ↓ ↓ I y I = e²ˣ ⁺ ᶜ

The exponential can be rewritten as a product. Ex.) e²ˣ ⁺ ᶜ ---> e²ˣ • eᶜ

Because C is a constant and "e" is also a number, C eats any constants that are added or multiplied to. So, eᶜ just becomes one large constant. (C) Also means that the + or - is not needed. Ex.) eᶜ ---> C

Use a point given, plug it in and solve for C.

A slope field shows the slope of a differential equation at certain vertical and horizontal intervals on the x-y plane, and can be used to determine the approximate tangent slope at a point on a curve, where the curve is some solution to the differential equation.

A solution curve is the curve that represents a solution (in the -plane). The slope field (or direction field) is a graphical representation of all short line segments through each point and with slope F ( x , y ) of a first-order differential equation.

Take the example of dy/dx = (x - y) at (3,4). - Here we see that dy/dx = (3−4) = −1 - So you would draw a line of slope −1 at (3,4). Repeat this for maybe 4 points to get a general slope field.

The Y coordinate

There are parallel segments horizontally - Ex.) dy/dx = y - 1

There are parallel segments vertically - Ex.) dy/dx = x + 1

That both X and Y is present in the differential equation - Ex.) dy/dx = x + y

parental functions

f(x) = x

f(x) = x²

f(x) = x³

f(x) = |x|

f(x) = 1/x

f(x) = eˣ

f(x) = log(x)

f(x) = √x

f(x) = sin(x)

f(x) = cos(x)

f(x) = tan(x)

The rate of change of a quantity (y) is directly proportional to the amount present. ↑ - Exponentially increasing.

dy/dt = ky dt = over time k = constant of proportionality

y = Ceᵏᵗ

To solve the problem, you must find k and t - To do so, plug in the first initial time (will be a pt.), then plug in the next time (will be a pt.) This will give you the C and K - Once you have C and K, plug in the values to the general equation to give you your general solution. By putting in the values into the equation, you will get the final value at a certain time.

y = Ceᵏᵗ - Similar to exponential growth, must find C and K and solve from there.

Mass or weight (typically given in grams) of the radioactive element.

time

The rate of change in the temperature of an object is directly proportional to the difference between the object's temperature and the temperature of the surrounding medium

A separable differential equation is any equation that can be written in the form y′=f(x)g(y). The method of separation of variables is used to find the general solution to a separable differential equation

dy/dx

Get the dy's and y's on one side and the dx's and x's on another, then integrate.

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