Answer:

General Formulas and Concepts:
Pre-Algebra
- Order of Operations
- Equality Properties
Algebra I
- Functions
- Function Notation
- Exponential Rule [Rewrite]:

Algebra II
- Natural logarithms ln and Euler's number e
Calculus
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]:
![\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx](https://img.qammunity.org/2022/formulas/mathematics/high-school/r5yh324r81plt97j3zrr5qi2xxczxlqi34.png)
U-Substitution
Logarithmic Integration:

Explanation:
*Note:
When solving differential equations in slope fields, disregard the integration constant C for variable y.
Step 1: Define


Step 2: Rewrite
Separation of Variables. Get differential equation to a form where we can integrate both sides and rewrite Leibniz Notation.
- [Separation of Variables] Rewrite Leibniz Notation:

- [Separation of Variables] Isolate y's together:

Step 3: Find General Solution Pt. 1
- [Differential] Integrate both sides:

- [dx Integral] Integrate [Integration Rule - Reverse Power Rule]:

Step 4: Find General Solution Pt. 2
Identify variables for u-substitution for dy.
- Set:

- Differentiate [Basic Power Rule]:

Step 5: Find General Solution Pt. 3
- [dy Integral] U-Substitution:

- [dy Integral] Integrate [Logarithmic Integration]:

- [Equality Property] e both sides:

- Simplify:

- Rewrite:

- Back-Substitute:

- [Equality Property] Isolate y:

General Form:

Step 6: Find Particular Solution
- Substitute in function values [General Form]:

- Simplify:

- [Equality Property] Isolate C:

- Rewrite:

- Substitute in C [General Form]:

∴ our particular solution is
.
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentials and Slope Fields
Book: College Calculus 10e