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Which of the following is the solution to the differential equation dy/dx=e^(y+x) with initial condition y(0) = -ln4
A) y= -x-ln4
B) y=x-ln4
C) y = -ln(-e^x+5)
D) y = -ln(e^x+3)
E) y = ln(e^x+3)

Respuesta :

Space

Answer:

C) y = -ln(-eˣ + 5)

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

  • Multiplication Property of Equality
  • Division Property of Equality
  • Addition Property of Equality
  • Subtraction Property of Equality

Algebra I

  • Function Notation
  • Exponential Rule [Multiplying]:                                                                       [tex]\displaystyle b^m \cdot b^n = b^{m + n}[/tex]
  • Exponential Rule [Rewrite]:                                                                            [tex]\displaystyle b^{-m} = \frac{1}{b^m}[/tex]

Algebra II

  • Log Properties
  • Natural log ln(x) and eˣ

Calculus

Antiderivatives - Integrals

Integration Constant C

U-Substitution

Slope Fields

  • Solving Differentials
  • Separation of Variables

Explanation:

Step 1: Define

[tex]\displaystyle \frac{dy}{dx} = e^{y + x} \\y(0) = -ln4[/tex]

Step 2: Rewrite

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

  1. [Differential Equation] Rewrite [Exponential Rule - Multiplying]:                 [tex]\displaystyle \frac{dy}{dx} = e^y \cdot e^x[/tex]
  2. [Diff Eq] Isolate x terms together [Multiplication Property of Equality]:       [tex]\displaystyle dy = e^y \cdot e^x dx[/tex]
  3. [Diff Eq] Isolate y terms together [Division Property of Equality]:                [tex]\displaystyle \frac{dy}{e^y} = e^x dx[/tex]
  4. [Diff Eq] Rewrite:                                                                                              [tex]\displaystyle \frac{1}{e^y} dy = e^x dx[/tex]
  5. [Diff Eq] Rewrite y [Exponential Rule - Rewrite]:                                           [tex]\displaystyle e^{-y} dy = e^x dx[/tex]

Step 3: Integrate Pt. 1

  1. [Diff Eq] Integrate both sides [Equality Property]:                                        [tex]\displaystyle \int {e^{-y}} \, dy = \int {e^x} \, dx[/tex]

Step 4: Identify Variables for U-Substitution

Set variables for u-sub for y.

u = -y

du = -dy

Step 5: Integrate Pt. 2

  1. [Integrals] Rewrite:                                                                                          [tex]\displaystyle -\int {-e^{-y}} \, dy = \int {e^x} \, dx[/tex]
  2. [Integrals] U-Substitution:                                                                               [tex]\displaystyle -\int {e^u} \, du = \int {e^x} \, dx[/tex]
  3. [Integrals] eˣ integration:                                                                             [tex]\displaystyle -e^u = e^x + C[/tex]
  4. [Integral Expression] Back-substitution:                                                        [tex]\displaystyle -e^{-y} = e^x + C[/tex]

Step 6: Solve Differential Equation Pt. 1

  1. [Int Exp] Divide -1 on both sides [Division Property of Equality]:                 [tex]\displaystyle e^{-y} = -e^x - C[/tex]
  2. [Int Exp] Natural log both sides (isolate y term) [Equality Property]:           [tex]\displaystyle -y = ln(-e^x - C)[/tex]        
  3. [Int Exp] Divide -1 on both sides [Division Property of Equality]:                 [tex]\displaystyle y = -ln(-e^x - C)[/tex]

This is our differential function.

Step 7: Solve Differential Equation Pt. 2

  1. [Diff Function] Substitute in given point:                                                       [tex]\displaystyle -ln4 = -ln(-e^0 - C)[/tex]
  2. [Diff Function] Evaluate exponent:                                                                [tex]\displaystyle -ln4 = -ln(-1 - C)[/tex]
  3. [Diff Function] Divide -1 on both sides [Division Property of Equality]:        [tex]\displaystyle ln4 = ln(-1 - C)[/tex]
  4. [Diff Function] e both sides [Equality Property]:                                            [tex]\displaystyle 4 = -1 - C[/tex]
  5. [Diff Function] Add 1 on both sides [Addition Property of Equality]:            [tex]\displaystyle 5 = -C[/tex]
  6. [Diff Function] Divide -1 on both sides [Division Property of Equality]:       [tex]\displaystyle -5 = C[/tex]
  7. [Diff Function] Rewrite:                                                                                  [tex]\displaystyle C = -5[/tex]
  8. [Diff Function] Substitute in Integration Constant C:                                  [tex]\displaystyle y = -ln(-e^x - -5)[/tex]
  9. [Diff Function] Simplify:                                                                                  [tex]\displaystyle y = -ln(-e^x + 5)[/tex]

Topic: Calculus

Unit: Slope Fields

Book: College Calculus 10e

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