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.)

General Formulas and Concepts:

Symbols

e (Euler's number) ≈ 2.7182

Pre-Algebra

Order of Operations: BPEMDAS

BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to Right  

Distributive Property

Equality Properties

Multiplication Property of EqualityDivision Property of EqualityAddition Property of EqualitySubtraction Property of Equality

Algebra I

Functions

Point-Slope Form: y - y₁ = m(x - x₁)

x₁ - x coordinatey₁ - y coordinatem - 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) = cxⁿf’(x) = c·nxⁿ⁻¹

Slope Fields

Solving Differentials - Integrals

Integration Constant C

U-Substitution

ln Integration:

Step-by-step explanation:

*Note:

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

Step 1: Define

x = 9, y = 7

Step 2: Rewrite Differential

Rewrite Leibniz Notation using Separation of Variables.

[Separation of Variables] Isolate x's together:                                             [Separation of Variables] Isolate y's together:                                            

Step 3: Integrate Pt. 1

Solving general form of differential using integration.

[Differential] Integrate both sides:                                                                

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

[Integrals] U-Substitution:                                                                               [Integrals] ln Integration:                                                                                Back-Substitute:                                                                                             [Equality Property] Raise e on both sides:                                                    Simplify:                                                                                                          

General Form:

Step 6: Solve Particular Solution

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

Substitute in variables [General Form]:                                                         [Particular] |Absolute Value| Subtract:                                                           [Particular] Evaluate absolute values:                                                         [Particular] Multiply:                                                                                        [Particular] Rewrite:                                                                                      

Substituting integration constant C into the general form:

Particular Solution (in Point-Slope Form):

Particular Solution (in Slope-Intercept Form):

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

Unit: Differentials and Slope Fields

Book: College Calculus 10e