General solution for differentiation

Discussion in 'Differentiation and Integration' started by Anfo, Feb 24, 2023.

  1. Anfo

    Anfo

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    Please can anyone help with these two equations.

    I am asked to provide the general solutions.

    a) 3^2 y^2dy/dx = 2 − 1

    b) 2dy/dx + 3 = −2 − 5

    Any help will be greatly appreciated!
     
    Anfo, Feb 24, 2023
    #1
  2. Anfo

    apprentus

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    a) To solve 3^2 y^2dy/dx = 2 − 1, we can first simplify it by dividing both sides by 3^2 y^2, which gives us:

    dy/dx = (2 - 1)/(3^2 y^2)

    dy/dx = 1/(9y^2)

    Now we can separate the variables by multiplying both sides by 9y^2 and dx:

    9y^2 dy = dx

    Next, we integrate both sides:

    ∫9y^2 dy = ∫dx

    3y^3 = x + C

    where C is the constant of integration.

    Therefore, the general solution to the differential equation is y = (x/3y^2) + C^(1/3).

    b) To solve 2dy/dx + 3 = −2 − 5, we can first simplify it by subtracting 3 from both sides and dividing by 2:

    dy/dx = -4/2

    dy/dx = -2

    Now we can integrate both sides with respect to x:

    ∫dy = ∫(-2)dx

    y = -2x + C

    where C is the constant of integration.

    Therefore, the general solution to the differential equation is y = -2x + C.
     
    apprentus, Mar 15, 2023
    #2
  3. Anfo

    HallsofIvy

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    a) 3^2 y^2dy/dx = 2 − 1
    It's strange that you would divide by 3^2 y^2= 9 y^2 and then, later, multiply by it!
    9y^2 dy/dx= 1
    9y^2dy= dx
    3y^3= x+ C
    I would NOT write it as y= a function of both x and y. Leave it as 3y^3= x+ C or
    write 3y^3- x= C.

    b) 2dy/dx + 3 = −2 − 5
    2 dy/dx+ 3= -7
    2 dy/dx= -10 (NOT -4!)
    dy= -5 dx
    y= -5x+ C.
     
    HallsofIvy, Aug 4, 2023
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