Verified Answer :

Space

Answer:

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

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]:                                                                  \(\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C\)

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:                                                                                            \(\displaystyle \int {\frac{1}{u}} \, dx = ln|u| + C\)

Explanation:

*Note:    

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

Step 1: Define

\(\displaystyle \frac{dy}{dx} = (2x - 1)y\)

\(\displaystyle y(1) = e\)

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:                                               \(\displaystyle \frac{1}{y} \ dy = (2x - 1) \ dx\)

Step 3: Find General Solution

  1. [Differential] Integrate both sides:                                                                   \(\displaystyle \int {\frac{1}{y}} \, dy = \int {(2x - 1)} \, dx\)
  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]:                 \(\displaystyle ln|y| = 2(\frac{x^2}{2}) - x + C\)
  6. Simplify:                                                                                                             \(\displaystyle ln|y| = x^2 - x + C\)
  7. [Equality Property] e both sides:                                                                     \(\displaystyle e^\bigg{ln|y|} = e^\bigg{x^2 - x + C}\)
  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

Related Questions