calc

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Find m and b so that y=mx+b is a solution to the differential equation
dy/dx=1/2x+y-1

  • calc -

    I will assume you mean (1/2)x and not 1/(2x)in the differential equation

    dy/dx = m, so we require that
    m = x/2 + mx+ b -1

    This is true if m = -1/2 and
    b = 1+m = 1/2

    y = -x/2 + 1/2 is a solution, but not the only solution, to the differential equation.

  • calc -

    dy/dx=(1/2)x + y - 1
    I assume that is what you mean.
    y = m x + b
    dy/dx = m
    so
    m = .5 x + mx + b - 1
    m = (.5 + m) x + b-1
    well if m is to be constant, m must be -.5
    that means
    -.5 = b-1
    b = .5
    so
    y = -.5 x + .5
    --------the end -----
    Now check with m = -.5 and b = +.5
    y = -.5 x + .5
    dy/dx = -.5

    dy/dx = .5 x + (-.5 x +.5) - 1
    = +.5 - 1
    = -.5
    check

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