Question

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)

Answer 1

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]:
  `\displaystyle b^m \cdot b^n = b^(m + n)`
• Exponential Rule [Rewrite]:
  `\displaystyle b^(-m) = (1)/(b^m)`

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

Step-by-step explanation:

Step 1: Define

`\displaystyle (dy)/(dx) = e^(y + x) \\y(0) = -ln4`

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]:
   `\displaystyle (dy)/(dx) = e^y \cdot e^x`
2. [Diff Eq] Isolate x terms together [Multiplication Property of Equality]:
   `\displaystyle dy = e^y \cdot e^x dx`
3. [Diff Eq] Isolate y terms together [Division Property of Equality]:
   `\displaystyle (dy)/(e^y) = e^x dx`
4. [Diff Eq] Rewrite:
   `\displaystyle (1)/(e^y) dy = e^x dx`
5. [Diff Eq] Rewrite y [Exponential Rule - Rewrite]:
   `\displaystyle e^(-y) dy = e^x dx`

Step 3: Integrate Pt. 1

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

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

Step 6: Solve Differential Equation Pt. 1

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

This is our differential function.

Step 7: Solve Differential Equation Pt. 2

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

Topic: Calculus

Unit: Slope Fields

Book: College Calculus 10e