Mathematics, 29.05.2021 06:10 shaakiraak

If dy/dx = ysec^2(x) and y= 5 when x = 0, then y=? А. e^tanx +4
B. e^tanx +5
C. 5e^tanx
D. tanx + 5
E. tanx + 5e^x

Answers: 2

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Answers

Answer from: hannahhoskings6989

C. $\displaystyle y = 5e^{tan(x)}$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to Right

Equality Properties

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

Algebra I

FunctionsFunction Notation|Absolute Values|Anything to the 0th power is 1

Algebra II

Logarithms and Natural logsEuler's number e

Calculus

Derivatives

Derivative Notation

Trig Derivatives

Differential Equations

Separation of variablesGeneral and particular solutions

Antiderivatives - Integration

Integration Constant C

Trig Integration

Logarithmic Integration

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \frac{dy}{dx} = ysec^2(x)$

x = 0, y = 5

Step 2: Rewrite Differential

Separation of variables

[Division Property of Equality] Isolate y terms together: $\displaystyle \frac{1}{y}\frac{dy}{dx} = sec^2(x)$ [Multiplication Property of Equality] Isolate x terms together: $\displaystyle \frac{1}{y}dy = sec^2(x)dx$

Step 3: Find General Solution

[Equality Property] Integrate both sides: $\displaystyle \int {\frac{1}{y}} \, dy = \int {sec^2(x)} \, dx$ [1st Integral] Integrate [Logarithmic Integration]: $\displaystyle ln|y| = \int {sec^2(x)} \, dx$ [2nd Integral] Integrate [Trig Integration]: $\displaystyle ln|y| = tan(x) + C$ [Equality Property] e both sides: $\displaystyle e^{ln|y|} = e^{tan(x) + C}$ Simplify: $\displaystyle |y| = Ce^{tan(x)}$ Rewrite: $\displaystyle y = \pm Ce^{tan(x)}$

Step 4: Find Particular Solution

Substitute in variables [Function]: $\displaystyle |5| = Ce^{tan(0)}$ Evaluate absolute value: $\displaystyle 5 = Ce^{tan(0)}$ Evaluate trig: $\displaystyle 5 = Ce^0$ Evaluate exponent: $\displaystyle 5 = C(1)$ Multiply: $\displaystyle 5 = C$ Rewrite: $\displaystyle C =5$ Substitute in C [General Solution]: $\displaystyle y = 5e^{tan(x)}$

∴ Our answer is C.

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

Unit: Differential Equations

Book: College Calculus 10e

Answer from: Quest

people went from manufacturing one object at a time by hand to manufacturing many objects in factories

Answer from: Quest

5050

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If dy/dx = ysec^2(x) and y= 5 when x = 0, then y=? А. e^tanx +4
B. e^tanx +5
C. 5e^tanx...

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