Solving First-Order Ordinary Differential Equations
Separable First-Order ODEs
- Separate the variables by putting all terms with “y” on one side of the equation and all terms with “x” on the other side.
- Integrate both sides of the equation with respect to their respective variables.
- Solve for y to find the general solution, adding a constant of integration at the end of the integration step; if an initial condition is given, substitute it to find the particular solution.
Key Steps in Detail
A separable first-order ODE can be written as dy/dx = f(x)g(y)
Read MoreDecision-Making Strategies: Risk, Certainty, and Uncertainty
Cooperatives grant members several rights, including participation in activities, access to necessary information, fulfillment of obligations, and the return of cooperative surpluses.
The Decision Matrix
A decision matrix is a table that encompasses all elements involved in decision-making. It provides a starting point for solving problems and offers a structured analysis:
- Strategies: Controllable variables representing alternatives or options to choose from.
- States of Nature: Uncontrollable variables
Understanding Quantitative Data Collection in Ecology
Quantitative data collection in the ecology of a habitat, community, or town involves interpreting collected information to draw conclusions.
Population statistics encompass all the data from which we wish to draw conclusions.
Statistical sample: A statistical sample is a randomly selected subset of the statistical population.
Statistics can be divided into descriptive analysis and statistical inference.
Descriptive analysis: This involves comments, graphics, tabular data, or numerical data that allow
Read MoreSolving Mixture and Differential Equations: Step-by-Step
Mixture Problems and Differential Equations
This document presents solutions to two types of mathematical problems: mixture problems and differential equations. The solutions are presented step-by-step, with explanations.
2.3.35 Mixture Problems
A tank contains 500L of water with 0.2 kg of salt. Initially, there are 5 kg of salt in the tank. The inflow and outflow rates are 5 L/min. We will determine the concentration at 10 minutes and after a leak develops, at 20 minutes.
a) Concentration at 10 minutes
Let
Read MoreProbability Problems: Discounts, Training, and Repairs
Sample Question (1) During its grand opening week, Stickler’s bicycle shop offers a “wheel of discount savings.” After customers select the items they wish to purchase, they spin the wheel to determine the discount they will receive. The wheel is divided into 12 equal slices. Six slices are red and award a 10% discount, three slices are white and award a 20% discount, and two slices are blue and award a 40% discount. The remaining slice is gold and awards a 100% discount! 20 10 10 10 100 40 10 10
Mastering Polynomial Multiplication: Rules and Examples
Rules of Exponents for Multiplication of Polynomials
Rule #1: Rule of Exponent 1
If a is a real number, then a = a1.
This rule states that when a variable has no exponent, its exponent is one (1).
Examples:
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- 51 = 5
- (-6)1 = -6
- (3a3b4)1 = 3a3b4
Rule #2: Multiplication of Powers
If a is any real number and m, n are positive integers, then
am · an = am + n
This means that when we multiply exponents with equal bases, we add the exponents.
Examples:
- x7 · x8 = x7 + 8 = x15
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- y4 · y6 = y4 + 6 = y10
(-4)2 · (-4) = (
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