Mathematics, 10.03.2021 21:20 shamiya15

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.)

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Answer from: angellong94

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

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: $\displaystyle \int {\frac{1}{x}} \, dx = ln|x| + C$

Step-by-step explanation:

*Note:

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

Step 1: Define

$\displaystyle \frac{dy}{dx} = \frac{y - 5}{x - 8}$

x = 9, y = 7

Step 2: Rewrite Differential

Rewrite Leibniz Notation using Separation of Variables.

[Separation of Variables] Isolate x's together: $\displaystyle dy = \frac{y - 5}{x - 8}dx$ [Separation of Variables] Isolate y's together: $\displaystyle \frac{1}{y - 5}dy = \frac{1}{x - 8}dx$

Step 3: Integrate Pt. 1

Solving general form of differential using integration.

[Differential] Integrate both sides: $\displaystyle \int {\frac{1}{y - 5}} \, dy = \int {\frac{1}{x - 8}} \, dx$

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

[Integrals] U-Substitution: $\displaystyle \int {\frac{1}{u}} \, du = \int {\frac{1}{z}} \, dz$ [Integrals] ln Integration: $\displaystyle ln|u| = ln|z| + C$ Back-Substitute: $\displaystyle ln|y - 5| = ln|x - 8| + C$ [Equality Property] Raise e on both sides: $\displaystyle e^{ln|y - 5|} = e^{ln|x - 8| + C}$ 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.

Substitute in variables [General Form]: $\displaystyle |7 - 5| = C|9 - 8|$ [Particular] |Absolute Value| Subtract: $\displaystyle |2| = C|1|$ [Particular] Evaluate absolute values: $\displaystyle 2 = C(1)$ [Particular] Multiply: $\displaystyle 2 = C$ [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): $\displaystyle y = 2x - 11$