Unit 07: Differential Equations

Question 1

Differential Equation

Answer 1

An equation that relates a function with its derivative, written in the form dy/dx = f(x, y).

Question 2

Slope Field

Answer 2

A graphical tool used to visualize solutions of a first-order differential equation consisting of a grid of short line segments representing slopes.

Question 3

Slope (at a point)

Answer 3

The value of the derivative dy/dx at a specific point, representing the rate of change of the function at that point.

Question 4

Initial Condition

Answer 4

A specific point (x₀, y₀) that the solution curve must pass through; used to find a particular solution.

Question 5

Particular Solution

Answer 5

A single function that satisfies the differential equation and passes through a specific given initial condition.

Question 6

Constant of Integration (C)

Answer 6

The constant added to an antiderivative to represent the family of all possible solutions (the general solution).

Question 7

Undefined Slope

Answer 7

Occurs when the DE has a zero in the denominator; graphically represented by no slope drawn or a vertical asymptote.

Question 8

Separation of Variables

Answer 8

A method to solve DEs by algebraically rearranging terms so that all y/dy terms are on one side and all x/dx terms are on the other.

Question 9

Leibniz Notation

Answer 9

The notation (dy/dx) that treats the derivative as a ratio of differentials, essential for the separation process.

Question 10

General Solution

Answer 10

The solution to a DE that includes an arbitrary constant C, representing a family of functions.

Question 11

Explicit Form

Answer 11

A solution written in the form y = f(x), where the dependent variable y is isolated on one side.

Question 12

Growth/Decay Differential Equation

Answer 12

An equation where the rate of change is proportional to the variable itself: dy/dt = ky.

Question 13

Exponential Growth and Decay Formula

Answer 13

The general solution to the growth/decay DE: y = Ce^(kt).

Question 14

Initial Amount (C)

Answer 14

The quantity present at time t = 0; the starting value.

Question 15

Growth or Decay Rate (k)

Answer 15

A constant representing the proportionality factor. k > 0 means growth; k < 0 means decay.

Question 16

Half-life

Answer 16

The amount of time (t) required for a quantity to fall to half of its initial value (y = 1/2 C).

Question 17

Constructing a Slope Field (Steps)

Answer 17

1. Select coordinates (x,y). 2. Substitute values into DE to calculate slope. 3. Draw a small line segment with that slope.

Question 18

Relationship between dy/dx and Graph Shape

Answer 18

> 0: segment slants upward; < 0: segment slants downward; = 0: segment is horizontal.

Question 19

Finding a Particular Solution Graphically

Answer 19

Start at the initial condition point and ‘follow the flow’ of the line segments in the slope field.

Question 20

Connection with Integrals

Answer 20

Solving a DE algebraically involves the integral: ∫(dy/dx) dx = ∫f(x) dx = F(x) + C.

Question 21

Euler’s Method

Answer 21

A numerical technique used to approximate a particular solution to a differential equation.

Question 22

Euler’s Method Formula (x-value)

Answer 22

xₙ = xₙ₋₁ + h

Question 23

Euler’s Method Formula (y-value)

Answer 23

yₙ = yₙ₋₁ + h · F(xₙ₋₁, yₙ₋₁)

Question 24

Step Size (h)

Answer 24

A small distance moved along the x-axis in Euler’s Method; smaller h leads to more accurate approximations.

Question 25

The Strategy for Separation (Steps)

Answer 25

1. Rewrite notation (y' to dy/dx). 2. Separate terms. 3. Integrate both sides. 4. Solve for y (isolate).

Question 26

Handling the Constant C in Separation

Answer 26

Constants from both sides are combined into a single '+C' on the x-side of the equation.

Question 27

Integration Techniques in Separation

Answer 27

Solving the x-side often requires advanced methods like u-substitution.

Question 28

Logarithmic Solutions

Answer 28

If integration results in ln|y| = f(x) + C, you must exponentiate both sides (base e) to solve for y.

Question 29

Isolating y: y² = f(x) + C

Answer 29

y = ±√(f(x) + C)

Question 30

Isolating y: y³ = f(x) + C

Answer 30

y = ∛(f(x) + C)

Question 31

Deriving Growth Formula: Exponentiation Step

Answer 31

e^(ln|y|) = e^(kt+C₁) becomes y = e^(kt) · e^(C₁).

Question 32

Properties of Exponents in Growth/Decay

Answer 32

The term e^C is just a constant and is replaced by the coefficient C in the final equation y = Ce^(kt).

Question 33

Finding C and k with Two Data Points

Answer 33

1. Create two equations: y₁ = Ce^(kt₁) and y₂ = Ce^(kt₂). 2. Isolate C in one. 3. Substitute into the second to solve for k. 4. Solve for C.