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Math Help - differentiation, separation of variables???

  1. #1
    Member i_zz_y_ill's Avatar
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    differentiation, separation of variables???

    the positive quantities x,y,z are related and vary with time t, where t is bigger or equal to 0. The value of x is described by the differential equation

    dx/dt +2x = t + 1

    t=0,x=1

    (i) solve to get x in terms of t (DONE)
    (ii)the quantity y is related to x by te D.e 2xdy/dx = y t=0,y=4
    solve to find y in terms of x,Henc express y in terms of t

    my working: (integral)1/y(dy) = (integral)1/2x(dx)
    lny = 0.5ln2x + C
    e^lny = e^((2x^0.5)+c)
    y = ae^(2x)^0.5

    in actual fact a = 4 and this is easily found if separate at start with
    (integral)2/y(dy)=(integral)1/x(dx)

    this gives 2lny = lnx +c
    y^2=e^lnx+c
    y^2=Ae^lnx
    y=Ax^0.5 initial conditions giving A=4

    My method gives basically lny = ln(2x)^0.5 + c
    y=Ae^ln(2x)^0.5
    y=A(2x)^0.5 initiial conditions giving A=4/root2

    If someone could point out my wrong working that would be really helpful ,whether its my initial separation or rule of logs im not sure thnx,
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  2. #2
    Moo
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    Hello,

    my working: (integral)1/y(dy) = (integral)1/2x(dx)
    lny = 0.5ln2x + C
    The problem is that \int \frac 1{2x} dx=\frac 12 \int \frac 1x dx=\frac 12 \cdot \ln x+c \neq \frac 12 \cdot \ln 2x+c
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