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  • Nov 14, 2010, 07:20 AM
    mayya
    Differential Equation Questions
    Q.1 Solve the given differential equation subject to the indicated initial condition.
    xy(1+xy^2)dy/dx = 1, y(1)=0

    Q.2 Find the orthogonal trajectories of the following family of curves.
    y^3+3x^2y=c1

    Q.3 Determine whether the functions f1(x)=x, f2(x)=x ln x, f3(x) = x^2lnx are linearly independent or linearly dependent on (0,∞).
  • Nov 14, 2010, 02:17 PM
    galactus
    Quote:

    Originally Posted by mayya View Post
    Q.1 Solve the given differential equation subject to the indicated initial condition.
    xy(1+xy^2)dy/dx = 1, y(1)=0

    Is there a possible typo? Just asking because this is really not doable by the usual means. I even ran it through Maple for a check and it gave me a crazy solution involving the Lambert W function. What section of DE did this come from? Bernoulli? Separable? Non-Homogeneous? Ricatti? Something else?

    Quote:

    Q.2 Find the orthogonal trajectories of the following family of curves.
    y^3+3x^2y=c1
    If f(x,y)=dy/dx is the DE of one family, then the DE for the orthogonal trajectories of the other family is dy/dx=-1/f(x,y).

    Suppose we have . Then, the DE is

    . Thus, the DE of the orthogonal family is



    It may be easier to find dx/dy rather than dy/dx.

    From the given function, by implicit differentiation w.r.t x, we get



    Now, find the orthogonal by using the info I gave.

    Quote:

    Q.3 Determine whether the functions f1(x)=x, f2(x)=x ln x, f3(x) = x^2lnx are linearly independent or linearly dependent on (0,∞).
    I can not read what interval you mean?

    But, the functions are linearly independent when neither is a constant multiple of the other on an interval.

    For instance,

    These are linearly dependent on the interval since can be written as a linear combo of



    You can also check the Wronskian. It is linearly independent if the Wronskian is not equal to 0 for every x in the interval.

    A Wronskian is the determinant of the matrix made up of the functions and their first and second derivatives.

    I get

    Does this equal 0 on the interval?
  • Nov 23, 2010, 12:46 AM
    mayya
    Comment on galactus's post
    Q.1 comes from Bernoulli section...
    And I can't type Q.3 in right way but it is in the same format that you have given. Looking for further help. Thanks.

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