Answer:
C) y = -ln(-eˣ + 5)
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to RightEquality Properties
Multiplication Property of EqualityDivision Property of EqualityAddition Property of EqualitySubtraction Property of EqualityAlgebra I
Function NotationExponential 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 PropertiesNatural log ln(x) and eˣCalculus
Antiderivatives - Integrals
Integration Constant C
U-Substitution
Slope Fields
Solving DifferentialsSeparation of VariablesExplanation:
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.
[Differential Equation] Rewrite [Exponential Rule - Multiplying]: [tex]\displaystyle \frac{dy}{dx} = e^y \cdot e^x[/tex][Diff Eq] Isolate x terms together [Multiplication Property of Equality]: [tex]\displaystyle dy = e^y \cdot e^x dx[/tex][Diff Eq] Isolate y terms together [Division Property of Equality]: [tex]\displaystyle \frac{dy}{e^y} = e^x dx[/tex][Diff Eq] Rewrite: [tex]\displaystyle \frac{1}{e^y} dy = e^x dx[/tex][Diff Eq] Rewrite y [Exponential Rule - Rewrite]: [tex]\displaystyle e^{-y} dy = e^x dx[/tex]Step 3: Integrate Pt. 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
[Integrals] Rewrite: [tex]\displaystyle -\int {-e^{-y}} \, dy = \int {e^x} \, dx[/tex][Integrals] U-Substitution: [tex]\displaystyle -\int {e^u} \, du = \int {e^x} \, dx[/tex][Integrals] eˣ integration: [tex]\displaystyle -e^u = e^x + C[/tex][Integral Expression] Back-substitution: [tex]\displaystyle -e^{-y} = e^x + C[/tex]Step 6: Solve Differential Equation Pt. 1
[Int Exp] Divide -1 on both sides [Division Property of Equality]: [tex]\displaystyle e^{-y} = -e^x - C[/tex][Int Exp] Natural log both sides (isolate y term) [Equality Property]: [tex]\displaystyle -y = ln(-e^x - C)[/tex] [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
[Diff Function] Substitute in given point: [tex]\displaystyle -ln4 = -ln(-e^0 - C)[/tex][Diff Function] Evaluate exponent: [tex]\displaystyle -ln4 = -ln(-1 - C)[/tex][Diff Function] Divide -1 on both sides [Division Property of Equality]: [tex]\displaystyle ln4 = ln(-1 - C)[/tex][Diff Function] e both sides [Equality Property]: [tex]\displaystyle 4 = -1 - C[/tex][Diff Function] Add 1 on both sides [Addition Property of Equality]: [tex]\displaystyle 5 = -C[/tex][Diff Function] Divide -1 on both sides [Division Property of Equality]: [tex]\displaystyle -5 = C[/tex][Diff Function] Rewrite: [tex]\displaystyle C = -5[/tex][Diff Function] Substitute in Integration Constant C: [tex]\displaystyle y = -ln(-e^x - -5)[/tex][Diff Function] Simplify: [tex]\displaystyle y = -ln(-e^x + 5)[/tex]Topic: Calculus
Unit: Slope Fields
Book: College Calculus 10e