Question

BC Calculus

If dy / dx = (2x - 1)y , y(1) = e, find y(2).

Answer 1

`\displaystyle y(2) = e^3`

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Equality Properties

Algebra I

• Functions
• Function Notation
• Exponential Rule [Multiplying]:
  `\displaystyle b^m \cdot b^n = b^(m + n)`

Algebra II

• Natural Logarithms ln and Euler's number e

Calculus

Derivatives

Derivative Notation

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Slope Fields

• Separation of Variables
• Solving Differentials

Integrals

• Antiderivatives

Integration Constant C

Integration Rule [Reverse Power Rule]:
`\displaystyle \int {x^n} \, dx = (x^(n + 1))/(n + 1) + C`

Integration Property [Multiplied Constant]:
`\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx`

Integration Property [Addition/Subtraction]:
`\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx`

Logarithmic Integration:
`\displaystyle \int {(1)/(u)} \, dx = ln|u| + C`

Step-by-step explanation:

*Note:

When solving differential equations in slope fields, disregard the integration constant C for variable y.

Step 1: Define

`\displaystyle (dy)/(dx) = (2x - 1)y`

`\displaystyle y(1) = e`

Step 2: Rewrite

Separation of Variables. Get differential equation to a form where we can integrate both sides and rewrite Leibniz Notation.

1. [Separation of Variables] Rewrite Leibniz Notation:
   `\displaystyle dy = (2x - 1)y \ dx`
2. [Separation of Variables] Isolate y's together:
   `\displaystyle (1)/(y) \ dy = (2x - 1) \ dx`

Step 3: Find General Solution

1. [Differential] Integrate both sides:
   `\displaystyle \int {(1)/(y)} \, dy = \int {(2x - 1)} \, dx`
2. [dy Integral] Integrate [Logarithmic Integration]:
   `\displaystyle ln|y| = \int {(2x - 1)} \, dx`
3. [dx Integral] Rewrite [Integration Property - Addition/Subtraction]:
   `\displaystyle ln|y| = \int {2x} \, dx - \int {} \, dx`
4. [1st dx Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle ln|y| = 2\int {x} \, dx - \int {} \, dx`
5. [dx Integrals] Integrate [Integration Rule - Reverse Power Rule]:
   `\displaystyle ln|y| = 2((x^2)/(2)) - x + C`
6. Simplify:
   `\displaystyle ln|y| = x^2 - x + C`
7. [Equality Property] e both sides:
   `\displaystyle e^\biggln = e^\bigg{x^2 - x + C}`
8. Simplify:
   `\displaystyle |y| = Ce^\bigg{x^2 - x}`
9. Rewrite:
   `\displaystyle y = \pm Ce^\bigg{x^2 - x}`

General Solution:
`\displaystyle y = \pm Ce^\bigg{x^2 - x}`

Step 4: Find Particular Solution

1. Substitute in function values [General Solution]:
   `\displaystyle e = \pm Ce^\bigg{1^2 - 1}`
2. Simplify:
   `\displaystyle e = \pm C`
3. Rewrite:
   `\displaystyle C = e`
4. Substitute in C [General Solution]:
   `\displaystyle y = e \bigg( e^\bigg{x^2 - x} \bigg)`
5. Simplify [Exponential Rule - Multiplying]:
   `\displaystyle y = e^\bigg{x^2 - x + 1}`

Particular Solution:
`\displaystyle y = e^\bigg{x^2 - x + 1}`

Step 5: Solve

1. Substitute in x [Particular Solution]:
   `\displaystyle y(2) = e^\bigg{2^2 - 2 + 1}`
2. Simplify:
   `\displaystyle y(2) = e^3`

∴ our final answer is
`\displaystyle y(2) = e^3`.

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentials and Slope Fields

Book: College Calculus 10e