Answers
Answer:
[tex]\displaystyle y = 2e^\bigg{\frac{x^3}{3}} + 1[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of OperationsEquality PropertiesAlgebra I
FunctionsFunction NotationExponential Rule [Rewrite]: [tex]\displaystyle b^{-m} = \frac{1}{b^m}[/tex]Algebra II
Natural logarithms ln and Euler's number eCalculus
Derivatives
Derivative Notation
Basic Power Rule:
f(x) = cxⁿ f’(x) = c·nxⁿ⁻¹Slope Fields
Separation of VariablesSolving DifferentialsIntegrals
AntiderivativesIntegration Constant C
Integration Rule [Reverse Power Rule]: [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Property [Addition/Subtraction]: [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]
U-Substitution
Logarithmic Integration: [tex]\displaystyle \int {\frac{1}{u}} \, dx = ln|u| + C[/tex]
Step-by-step explanation:
*Note:
When solving differential equations in slope fields, disregard the integration constant C for variable y.
Step 1: Define
[tex]\displaystyle \frac{dy}{dx} = x^2(y - 1)[/tex]
[tex]\displaystyle f(0) = 3[/tex]
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: [tex]\displaystyle dy = x^2(y - 1) \ dx[/tex][Separation of Variables] Isolate y's together: [tex]\displaystyle \frac{1}{y - 1} \ dy = x^2 \ dx[/tex]Step 3: Find General Solution Pt. 1
[Differential] Integrate both sides: [tex]\displaystyle \int {\frac{1}{y - 1}} \, dy = \int {x^2} \, dx[/tex][dx Integral] Integrate [Integration Rule - Reverse Power Rule]: [tex]\displaystyle \int {\frac{1}{y - 1}} \, dy = \frac{x^3}{3} + C[/tex]Step 4: Find General Solution Pt. 2
Identify variables for u-substitution for dy.
Set: [tex]\displaystyle u = y - 1[/tex]Differentiate [Basic Power Rule]: [tex]\displaystyle du = dy[/tex]Step 5: Find General Solution Pt. 3
[dy Integral] U-Substitution: [tex]\displaystyle \int {\frac{1}{u}} \, du = \frac{x^3}{3} + C[/tex][dy Integral] Integrate [Logarithmic Integration]: [tex]\displaystyle ln|u| = \frac{x^3}{3} + C[/tex][Equality Property] e both sides: [tex]\displaystyle e^\bigg{ln|u|} = e^\bigg{\frac{x^3}{3} + C}[/tex]Simplify: [tex]\displaystyle |u| = Ce^\bigg{\frac{x^3}{3}}[/tex]Rewrite: [tex]\displaystyle u = \pm Ce^\bigg{\frac{x^3}{3}}[/tex]Back-Substitute: [tex]\displaystyle y - 1 = \pm Ce^\bigg{\frac{x^3}{3}}[/tex][Equality Property] Isolate y: [tex]\displaystyle y = \pm Ce^\bigg{\frac{x^3}{3}} + 1[/tex]General Form: [tex]\displaystyle y = \pm Ce^\bigg{\frac{x^3}{3}} + 1[/tex]
Step 6: Find Particular Solution
Substitute in function values [General Form]: [tex]\displaystyle 3 = \pm Ce^\bigg{\frac{0^3}{3}} + 1[/tex]Simplify: [tex]\displaystyle 3 = \pm C + 1[/tex][Equality Property] Isolate C: [tex]\displaystyle 2 = \pm C[/tex]Rewrite: [tex]\displaystyle C = 2[/tex]Substitute in C [General Form]: [tex]\displaystyle y = 2e^\bigg{\frac{x^3}{3}} + 1[/tex]∴ our particular solution is [tex]\displaystyle y = 2e^\bigg{\frac{x^3}{3}} + 1[/tex].
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentials and Slope Fields
Book: College Calculus 10e
Related Questions
what is the value of the expression 2x + 5y when x = 3 and y = 2
Answers
Answer:
16
Step-by-step explanation:
If x = 3 and , y = 2.
2 x 3 = 6 and 5 x 2 = 10.
10 + 6 = 16
The value of the expression 2x + 5y when x = 3 and y = 2 is 16.
We have to determine, the value of the expression 2x + 5y when x = 3 and y = 2.
According to the question,
Expression; 2x + 5y
To determine the value of the expression calculation must be done in a single unit following all the steps given below.
The value of expression 2x+5y,
When the value of x = 3 and y = 2,
Therefore,
[tex]= 2x + 5y[/tex]
Substitute the value of x = 3 and y = 2 in the given expression,
[tex]= 2(3)+ 5(2)\\\\= 6 + 10\\\\= 16[/tex]
Hence, The value of the expression 2x + 5y when x = 3 and y = 2 is 16.
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Answer:
the answer is (X,Y)=(4,5)
For sure
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11/20
Step-by-step explanation:
do BEDMAS
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Answer:
Answer is 1/3
Step-by-step explanation:
If you divide the numerator and denominator by five you get 3/9
3/9 can be reduced if you divide the numerator and denominator by 3, then your answer is 1/3 (since 3 goes into 3 once, and 3 goes into 9 three times.)
X^2+12=40 what number must you add to complete the square?
Answers
Answer:
Step-by-step explanation:
1) If you really meant X^2+12=40, then this simplifies to x^2 = 28, and therefore x = ±√28, or x = ±2√7.
2) If you meant X^2+12x=40:
a) Take half of the coefficient of x: that would be (1/2)(12), or 6.
b) Square this result, obtaining: 6² = 36
c) Add this 36 to x^2 + 12x + 40, and then subtract it: We get:
x² + 12x + 36 - 36 = 40, or x² + 12x + 36 = 76
We have to add 36, as indicated above, to "complete the square."
What is the quotient of 4.8 divided by -0.4
Answers
Answer:
the quotient is keeping batman alive
Answer:
A-(4÷-0.4)+(0.8)
b-(4÷-0.4)+(0.8÷-0.4)
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Answers
Answer:
lol same
Step-by-step explanation:
idek
Answer:
What grade are you in? You aren't supposed to do all your missing work in one day try to get onto your work currently if you have extra time do your missing work. Remember you wanna do good on it you don't wanna finish all and be stressed out because you'll have even more missing work. If you do it just to finish it won't help either. Be focused while doing it but also be free and don't bother yourself. Such as if you played valorant for 5 hours do a missing assignment then keep on playing (that is if you already finished what you did)
Step-by-step explanation:
Rotate point C(6,-3) 90 clockwise
PLEASE HELP
Answers
The answer is (-3, -6)
Step-by-step explanation:
turn the x into a y, and the y into a opposite x.
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Answers
Answer:
C) 88.81 dB
Step-by-step explanation:
Answer: c. 88.81
Step-by-step explanation:
took the test
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33 is what percent of 40?
Answers
Answer:
33 is 83% of 40.
Step-by-step explanation:
33/40 = 0.825
0.825 equals 82.5%
82.5% rounded is 83%.
I hope this helps!
The required value of percentage is 82.5%. Here, 33 represents approximately 82.5 percent of the total value of 40.
To find the percent, we need to determine what portion of 40 is represented by 33. We can set up a proportion to solve this problem.
Let x represent the unknown percentage.
We can set up the proportion:
33/40 = x/100
To solve for x, we can cross-multiply and then divide:
= (33 × 100) / 40
= 3300/40
Apply division operation to get:
= 82.5
Therefore, 33 is 82.5 percent of 40.
This means that 33 is equivalent to 82.5 percent of 40.
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brainly.com/question/24159063
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Nina delivers newspapers. In the first week, she delivers 8 papers. In the second week, she delivers 32 papers. Which equation can Nina use to find how many times as many papers she delivered the second week as the first week?
Answers
Answer:
a) The equation can Nina use to find how many times as many papers she delivered the second week as the first week is given as:
8x = 32
b) Nina delivered 4 times as many papers the second week as the first week
Step-by-step explanation:
From the question, we are told that:
Nina delivers newspapers.
In the first week, she delivers 8 papers.
In the second week, she delivers 32 papers.
Let x represent the number of times
The equation can Nina use to find how many times as many papers she delivered the second week as the first week is given as:
8paper × x = 32 papers
8x = 32
Solving for x
8x = 32
x = 32/8
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promiscious girl
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Answers
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Answers
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Step-by-step explanation:
btw "^" means that whatever number following it is an exponent.
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Answer:
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Step-by-step explanation
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Step-by-step explanation:
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Step-by-step explanation:
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Answer:
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Step-by-step explanation:
Answer:
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Step-by-step explanation:
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NEEXDD QUICK ANSWER TIMED TEST
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