Answer:
General Formulas and Concepts:
<u>Pre-Algebra</u>
- Order of Operations
- Equality Properties
<u>Algebra I</u>
- Functions
- Function Notation
- Exponential Rule [Rewrite]:
<u>Algebra II</u>
- Natural logarithms ln and Euler's number e
<u>Calculus</u>
Derivatives
Derivative Notation
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Slope Fields
Integrals
Integration Constant C
Integration Rule [Reverse Power Rule]:
Integration Property [Addition/Subtraction]:
U-Substitution
Logarithmic Integration:
Step-by-step explanation:
*Note:
When solving differential equations in slope fields, disregard the integration constant C for variable y.
<u />
<u>Step 1: Define</u>
<u>Step 2: Rewrite</u>
<em>Separation of Variables. Get differential equation to a form where we can integrate both sides and rewrite Leibniz Notation.</em>
- [Separation of Variables] Rewrite Leibniz Notation:
- [Separation of Variables] Isolate <em>y</em>'s together:
<u>Step 3: Find General Solution Pt. 1</u>
- [Differential] Integrate both sides:
- [dx Integral] Integrate [Integration Rule - Reverse Power Rule]:
<u>Step 4: Find General Solution Pt. 2</u>
<em>Identify variables for u-substitution for dy.</em>
- Set:
- Differentiate [Basic Power Rule]:
<u>Step 5: Find General Solution Pt. 3</u>
- [dy Integral] U-Substitution:
- [dy Integral] Integrate [Logarithmic Integration]:
- [Equality Property] e both sides:
- Simplify:
- Rewrite:
- Back-Substitute:
- [Equality Property] Isolate <em>y</em>:
General Form:
<u>Step 6: Find Particular Solution</u>
- Substitute in function values [General Form]:
- Simplify:
- [Equality Property] Isolate <em>C</em>:
- Rewrite:
- Substitute in <em>C</em> [General Form]:
∴ our particular solution is .
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentials and Slope Fields
Book: College Calculus 10e