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Math Help - pde question quasi lineer equation

  1. #1
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    pde question quasi lineer equation

    QUESTİON:
    x(y^2+z)z_x - y(x^2+z)z_y =(x^2-y^2).z
    thanks for your helps dear friends.
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  2. #2
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    Quote Originally Posted by sah_mat View Post
    QUESTİON:
    x(y^2+z)z_x - y(x^2+z)z_y =(x^2-y^2).z
    thanks for your helps dear friends.
    The characteristic equations are

    \frac{dx}{x(y^2+z)} = \frac{dy}{-y(x^2+z)} = \frac{dz}{(x^2- y^2)z}<br />

    from which we can deduce

    \frac{dx}{x} + \frac{dy}{y} + \frac{dz}{z} = \;\;\; \Rightarrow\;\;\; xyz = c_1

    and

    x\, dx + y\, dy - dz = 0 \;\;\; \Rightarrow\;\;\; x^2 + y^2 -2 z = c_2

    giving the solution as

    x^2+y^2-2z = f(xyz)
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  3. #3
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    thanks for your solution,but one point ı am with trouble i did not understand your deduction which you did at here \frac{dx}{x(y^2+z)} = \frac{dy}{-y(x^2+z)} = \frac{dz}{(x^2- y^2)z}

    to get this \frac{dx}{x} + \frac{dy}{y} + \frac{dz}{z} =
    if you explain a little more ı will be gladfull to you.
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  4. #4
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    Quote Originally Posted by sah_mat View Post
    thanks for your solution,but one point ı am with trouble i did not understand your deduction which you did at here \frac{dx}{x(y^2+z)} = \frac{dy}{-y(x^2+z)} = \frac{dz}{(x^2- y^2)z}

    to get this \frac{dx}{x} + \frac{dy}{y} + \frac{dz}{z} =
    if you explain a little more ı will be gladfull to you.
    Sure, let

    \frac{dx}{x(y^2+z)} = \frac{dy}{-y(x^2+z)} = \frac{dz}{(x^2- y^2)z}

    or

    \frac{dx}{dt} =x(y^2+z),\;\; \frac{dy}{dt} = -y(x^2+z),\;\; \frac{dz}{dt} = (x^2- y^2)z

    so

    \frac{1}{x}\,\frac{dx}{dt} =y^2+z,\;\; \frac{1}{y}\, \frac{dy}{dt} = -(x^2+z),\;\; \frac{1}{z}\, \frac{dz}{dt} = x^2- y^2

    and if you add

    \frac{1}{x}\,\frac{dx}{dt} + \frac{1}{y}\,\frac{dy}{dt} + \frac{1}{x}\,\frac{dx}{dt} = 0

    or what I wrote earlier.
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  5. #5
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    i discovered that u are solving wşth a different way from my teacher taught me,and i am at the start of pde hence i have some misunderstandable points left about your solution if you let me to ask you i will be happy,
    how did you found c1 and c2 please explain me all steps ,than i will be capable of solving other questions with your way which is more understandable and easy to understand.thanks a lot for oyur helps danny.
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  6. #6
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    Quote Originally Posted by sah_mat View Post
    i discovered that u are solving wşth a different way from my teacher taught me,and i am at the start of pde hence i have some misunderstandable points left about your solution if you let me to ask you i will be happy,
    how did you found c1 and c2 please explain me all steps ,than i will be capable of solving other questions with your way which is more understandable and easy to understand.thanks a lot for oyur helps danny.
    Before I can help, we need to find common ground. Please solve the following so I ca nsee what your teacher taught you

    x z_x + y z_y = 0,

    x z_x +(x+y) z_y = z,

     z_x + z z_y = x.
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