Question

Need help please its Calculus. Ill give the 5 stars as well.

Answer 1

`\displaystyle y = 2e^\bigg{(x^3)/(3)} + 1`

General Formulas and Concepts:

Pre-Algebra

• Order of Operations
• Equality Properties

Algebra I

• Functions
• Function Notation
• Exponential Rule [Rewrite]:
  `\displaystyle b^(-m) = (1)/(b^m)`

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

U-Substitution

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

Explanation:

*Note:

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

Step 1: Define

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

`\displaystyle f(0) = 3`

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

Step 3: Find General Solution Pt. 1

1. [Differential] Integrate both sides:
   `\displaystyle \int {(1)/(y - 1)} \, dy = \int {x^2} \, dx`
2. [dx Integral] Integrate [Integration Rule - Reverse Power Rule]:
   `\displaystyle \int {(1)/(y - 1)} \, dy = (x^3)/(3) + C`

Step 4: Find General Solution Pt. 2

Identify variables for u-substitution for dy.

1. Set:
   `\displaystyle u = y - 1`
2. Differentiate [Basic Power Rule]:
   `\displaystyle du = dy`

Step 5: Find General Solution Pt. 3

1. [dy Integral] U-Substitution:
   `\displaystyle \int {(1)/(u)} \, du = (x^3)/(3) + C`
2. [dy Integral] Integrate [Logarithmic Integration]:
   `\displaystyle ln|u| = (x^3)/(3) + C`
3. [Equality Property] e both sides:
   `\displaystyle e^\biggu = e^\bigg{(x^3)/(3) + C}`
4. Simplify:
   `\displaystyle |u| = Ce^\bigg{(x^3)/(3)}`
5. Rewrite:
   `\displaystyle u = \pm Ce^\bigg{(x^3)/(3)}`
6. Back-Substitute:
   `\displaystyle y - 1 = \pm Ce^\bigg{(x^3)/(3)}`
7. [Equality Property] Isolate y:
   `\displaystyle y = \pm Ce^\bigg{(x^3)/(3)} + 1`

General Form:
`\displaystyle y = \pm Ce^\bigg{(x^3)/(3)} + 1`

Step 6: Find Particular Solution

1. Substitute in function values [General Form]:
   `\displaystyle 3 = \pm Ce^\bigg{(0^3)/(3)} + 1`
2. Simplify:
   `\displaystyle 3 = \pm C + 1`
3. [Equality Property] Isolate C:
   `\displaystyle 2 = \pm C`
4. Rewrite:
   `\displaystyle C = 2`
5. Substitute in C [General Form]:
   `\displaystyle y = 2e^\bigg{(x^3)/(3)} + 1`

∴ our particular solution is
`\displaystyle y = 2e^\bigg{(x^3)/(3)} + 1`.

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

Unit: Differentials and Slope Fields

Book: College Calculus 10e