Let y=f(x) be the particular solution to the differential equation dy/dx=2x−1/y^2 with the initial condition y(0)=3. Which of the following is an expression for f(x)?
A. √(x^2−x+9)

B. −3/3x^2−3x−1

C. (3x^2−3x)^1/3+3

D. (3x^2−3x+27)^1/3

Answers

Answer 1

Answer:

D. (3x^2-3x+27)^1/3

Step-by-step explanation:

dy/dx=2x−1/y^2

Cross multiply to get Y's and X's on different sides.

(y^2)dy = (2x-1)dx

Take the integral of each side

∫(y^2)dy = ∫(2x-1)dx

y^3/3 = 2x^2/2 - x + C

Multiply both sides by 3

y^3 = 3x^2 - 3x + C

Plug in (0,3) and solve for C

27 = 0 - 0 + C

C = 27

Plug in C to the equation

y^3 = 3x^2-3x+27

Cube root both sides to get just y

y = √(3x^2-3x+27)

y = (3x^2-3x+27)^1/3

D.

Answer 2

The particular solution of the equation will be "[tex](3x^2-3x+27)^\frac{1}{3}[/tex]".

Differential equation

According to the question,

→    [tex]\frac{dy}{dx}[/tex] = [tex]\frac{2x-1}{y^2}[/tex]

By applying cross-multiplication,

(y²)dy = (2x - 1)dx

By integrating both sides, we get

→ ∫(y²) dy = ∫(2x - 1) dx

           [tex]\frac{y^3}{3}[/tex] = [tex]\frac{2x^2}{2}[/tex] - x + C

By multiplying "3" both sides, we get

           y³ = 3x² - 3x + c

By substituting "(0,3)" in above equation,

          27 = 0 - 0 + C

            C = 27

By substituting value of C in the above equation,

         y³ = 3x² - 3x + 27

By cubing both sides,

y = √(3x²-3x+27)

  = [tex](3x^2-3x+27)^\frac{1}{3}[/tex]        

Thus the above answer i.e., "Option D" is correct.

Find out more information about differentiation here:

https://brainly.com/question/25125500


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Answers

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Answer:

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Step-by-step explanation:

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Step-by-step explanation:

Answer:

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Step-by-step explanation:

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Step-by-step explanation:

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Answers

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Answer:

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Step-by-step explanation:

hey there

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Answers

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Step-by-step explanation:

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Answers

Answer:

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Step-by-step explanation:

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Step-by-step explanation:

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EX:
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Step-by-step explanation:

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Question

The data in this question will be utilized for the next three questions. The number of pets a group of students has, are given below.

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Number of students:  1        2      1        5       6         2           3

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Step-by-step explanation:

Given

The above data

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Answers

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Answers

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