Question

Find the particular solution of the differential equation dy/dx=y−5/x−8 using the given boundary condition: y=7 when x=9.

(Use symbolic notation and fractions where needed.)

Answer 1

`\displaystyle y - 5 = 2(x - 8)`

General Formulas and Concepts:

Symbols

• e (Euler's number) ≈ 2.7182

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Distributive Property

Equality Properties

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

Algebra I

Functions

Point-Slope Form: y - y₁ = m(x - x₁)

• x₁ - x coordinate
• y₁ - y coordinate
• m - slope

Slope-Intercept Form: y = mx + b

• m - slope
• b - y-intercept

Algebra II

• Logarithms - ln and e

Calculus

Derivatives

Derivative Notation

Derivative of a constant is 0

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Slope Fields

Solving Differentials - Integrals

Integration Constant C

U-Substitution

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

Explanation:

*Note:

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

Step 1: Define

`\displaystyle (dy)/(dx) = (y - 5)/(x - 8)`

x = 9, y = 7

Step 2: Rewrite Differential

Rewrite Leibniz Notation using Separation of Variables.

1. [Separation of Variables] Isolate x's together:
   `\displaystyle dy = (y - 5)/(x - 8)dx`
2. [Separation of Variables] Isolate y's together:
   `\displaystyle (1)/(y - 5)dy = (1)/(x - 8)dx`

Step 3: Integrate Pt. 1

Solving general form of differential using integration.

1. [Differential] Integrate both sides:
   `\displaystyle \int {(1)/(y - 5)} \, dy = \int {(1)/(x - 8)} \, dx`

Step 4: Identify Variables

Set up u-substitution for right integral.

Integral w/ respect to y

u = y - 5

du = dy

Integral w/ respect to x

z = x - 8

dz = dx

Step 5: Integrate Pt. 2

1. [Integrals] U-Substitution:
   `\displaystyle \int {(1)/(u)} \, du = \int {(1)/(z)} \, dz`
2. [Integrals] ln Integration:
   `\displaystyle ln|u| = ln|z| + C`
3. Back-Substitute:
   `\displaystyle ln|y - 5| = ln|x - 8| + C`
4. [Equality Property] Raise e on both sides:
   `\displaystyle e^(ln|y - 5|) = e^(ln|x - 8| + C)`
5. Simplify:
   `\displaystyle |y - 5| = C|x - 8|`

General Form:
`\displaystyle |y - 5| = C|x - 8|`

Step 6: Solve Particular Solution

Since both sides have absolute value, assume that the particular solution will be positive.

1. Substitute in variables [General Form]:
   `\displaystyle |7 - 5| = C|9 - 8|`
2. [Particular] |Absolute Value| Subtract:
   `\displaystyle |2| = C|1|`
3. [Particular] Evaluate absolute values:
   `\displaystyle 2 = C(1)`
4. [Particular] Multiply:
   `\displaystyle 2 = C`
5. [Particular] Rewrite:
   `\displaystyle C = 2`

Substituting integration constant C into the general form:

Particular Solution (in Point-Slope Form):
`\displaystyle y - 5 = 2(x - 8)`

Particular Solution (in Slope-Intercept Form):
`\displaystyle y = 2x - 11`

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

Unit: Differentials and Slope Fields

Book: College Calculus 10e