Advanced Placement (AP), 31.08.2019 04:00 gamboaserg

Bc calculus
if dy / dx = (2x - 1)y , y(1) = e, find y(2).

Answers: 1

Show answers

Answers

Answer from: hasshh

$\displaystyle y(2) = e^3$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to Right

Equality Properties

Algebra I

FunctionsFunction NotationExponential Rule [Multiplying]: $\displaystyle b^m \cdot b^n = b^{m + n}$

Algebra II

Natural Logarithms ln and Euler's number e

Calculus

Derivatives

Derivative Notation

Basic Power Rule:

f(x) = cxⁿ f’(x) = c·nxⁿ⁻¹

Slope Fields

Separation of VariablesSolving Differentials

Integrals

Antiderivatives

Integration Constant C

Integration Rule [Reverse Power Rule]: $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

Integration Property [Multiplied Constant]: $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

Integration Property [Addition/Subtraction]: $\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx$

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

Explanation:

*Note:

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

Step 1: Define

$\displaystyle \frac{dy}{dx} = (2x - 1)y$

$\displaystyle y(1) = e$

Step 2: Rewrite

Separation of Variables. Get differential equation to a form where we can integrate both sides and rewrite Leibniz Notation.

[Separation of Variables] Rewrite Leibniz Notation: $\displaystyle dy = (2x - 1)y \ dx$ [Separation of Variables] Isolate y's together: $\displaystyle \frac{1}{y} \ dy = (2x - 1) \ dx$

Step 3: Find General Solution

[Differential] Integrate both sides: $\displaystyle \int {\frac{1}{y}} \, dy = \int {(2x - 1)} \, dx$ [dy Integral] Integrate [Logarithmic Integration]: $\displaystyle ln|y| = \int {(2x - 1)} \, dx$ [dx Integral] Rewrite [Integration Property - Addition/Subtraction]: $\displaystyle ln|y| = \int {2x} \, dx - \int {} \, dx$ [1st dx Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle ln|y| = 2\int {x} \, dx - \int {} \, dx$ [dx Integrals] Integrate [Integration Rule - Reverse Power Rule]: $\displaystyle ln|y| = 2(\frac{x^2}{2}) - x + C$ Simplify: $\displaystyle ln|y| = x^2 - x + C$ [Equality Property] e both sides: $\displaystyle e^\bigg{ln|y|} = e^\bigg{x^2 - x + C}$ Simplify: $\displaystyle |y| = Ce^\bigg{x^2 - x}$ Rewrite: $\displaystyle y = \pm Ce^\bigg{x^2 - x}$

General Solution: $\displaystyle y = \pm Ce^\bigg{x^2 - x}$

Step 4: Find Particular Solution

Substitute in function values [General Solution]: $\displaystyle e = \pm Ce^\bigg{1^2 - 1}$ Simplify: $\displaystyle e = \pm C$ Rewrite: $\displaystyle C = e$ Substitute in C [General Solution]: $\displaystyle y = e \bigg( e^\bigg{x^2 - x} \bigg)$ Simplify [Exponential Rule - Multiplying]: $\displaystyle y = e^\bigg{x^2 - x + 1}$

Particular Solution: $\displaystyle y = e^\bigg{x^2 - x + 1}$

Step 5: Solve

Substitute in x [Particular Solution]: $\displaystyle y(2) = e^\bigg{2^2 - 2 + 1}$ Simplify: $\displaystyle y(2) = e^3$

∴ our final answer is $\displaystyle y(2) = e^3$ .

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

Unit: Differentials and Slope Fields

Book: College Calculus 10e

Answer from: Quest

answer: cricket

explanation:

i know this because i play cricket.

give brainliest

Answer from: Quest

answer:

i would rather know when i would die.

Another question on Advanced Placement (AP)

Advanced Placement (AP), 24.06.2019 17:50

For review 1. in terms of which government is sovereign, differentiate among a unitary system, a confed- eral system, and a federal system of government. 2. to which level of government does the constitution distribute the enumerated po implied powers? concurrent powers? reserved powers? provide several power. 3. what matters fall within the scope of state sovereignty? differentiate among dual federalism, cooperative federalism, centralized federalism, and conflicted federalism. how does the national government use grants-in-aid, mandates, and preemption to direct the policy of state and local governments? 6. what do we mean by intergovernmental relations? why is the term a good description of u.s. federalism today? 7. list several advantages and several disadvantages of a federal system.

Answers: 2

Answer

Advanced Placement (AP), 25.06.2019 15:30

Which of the following types of stars had the smallest initial mass? which of the following types of stars had the smallest initial mass?

Answers: 1

Answer

Advanced Placement (AP), 25.06.2019 19:30

What did louis xiv do that create absolute monarchies?

Answers: 1

Answer

Advanced Placement (AP), 25.06.2019 23:20

You find a mysterious, dull substance on the ground and pick it up for a closer look. it is powdery, very light, and soft to the touch. the substance is most likely i. an organic mineral ii. a metallic mineral iii. a nonmetallic mineral i only ii only iii only i and ii i and iii

Answers: 1

Answer

You know the right answer?

Bc calculus
if dy / dx = (2x - 1)y , y(1) = e, find y(2)....

Questions