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Math Help - [SOLVED] Differentiation in explicit form

  1. #1
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    [SOLVED] Differentiation in explicit form

    Find the general solution of the differential equation

    y′ = \frac{2x^2+y^2}{xy}. in explicit form.
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  2. #2
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    Quote Originally Posted by ronaldo_07 View Post
    Find the general solution of the differential equation

    y′ = \frac{2x^2+y^2}{xy}. in explicit form.
    Use a substitution of  u = \frac{y}{x}

     y = xu

     \frac{dy}{dx} = u + x\frac{du}{dx}

     \frac{dy}{dx} = \frac{2x^2}{xy}+\frac{y^2}{xy}

     \frac{dy}{dx} = \frac{2x}{y}+\frac{y}{x}

     u + x\frac{du}{dx} = \frac{2}{u}+u

     x\frac{du}{dx} = \frac{2}{u}

     \frac{du}{u} = 2 \times \frac{dx}{x}
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  3. #3
    Member ronaldo_07's Avatar
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    What are we solving for in this question?
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    Quote Originally Posted by ronaldo_07 View Post
    What are we solving for in this question?
    Solve for u.

    Get it in the form:

     u = \text{ something }

    Then substitute back in for y. You know that  u = \frac{y}{x} . Hence:

     \frac{y}{x} = \text{ something }

     y = \text{ something } \times x
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  5. #5
    Member ronaldo_07's Avatar
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    Quote Originally Posted by Mush View Post
    Use a substitution of  u = \frac{y}{x}

     y = xu

     \frac{dy}{dx} = u + x\frac{du}{dx}

     \frac{dy}{dx} = \frac{2x^2}{xy}+\frac{y^2}{xy}

     \frac{dy}{dx} = \frac{2x}{y}+\frac{y}{x}

     u + x\frac{du}{dx} = \frac{2}{u}+u

     x\frac{du}{dx} = \frac{2}{u}

     \frac{du}{u} = 2 \times \frac{dx}{x}
    This is all a bit confusing to me. To find y do you not need to integrate somewhere?
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  6. #6
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    Quote Originally Posted by ronaldo_07 View Post
    This is all a bit confusing to me. To find y do you not need to integrate somewhere?
    Read post #4 more carefully.
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  7. #7
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    Quote Originally Posted by ronaldo_07 View Post
    This is all a bit confusing to me. To find y do you not need to integrate somewhere?
    At the very end of my first post you must apply integrated with respect to u on the LHS, and with respect to x on the RHS. Then go to my 2nd post which tells you how to proceed to change back from u to y.
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