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Math Help - How to solve this second order Diff. Equation.

  1. #1
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    How to solve this second order Diff. Equation.

    I took a course on ODE last year, but I do not remember seeing something of this sort:

    (d^2V/dx^2)= N/sqrt(V) Here V is in terms of x, and N is a constant.

    Thank u for any suggestions!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: How to solve this second order Diff. Equation.

    Being F(x) an anti-derivative of \frac{N}{\sqrt{V(x)}} then, V=C_1x+\int F(x)\;dx .
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    Re: How to solve this second order Diff. Equation.

    Not so simple... (see attachment)
    Attached Thumbnails Attached Thumbnails How to solve this second order Diff. Equation.-nonlinear-edo.jpg  
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    Re: How to solve this second order Diff. Equation.

    I don't agree with the method proposed by FernandoRevilla because F(x) is defined relatively to V(x) which is not known. i.e. the solution is expressed as a function of the solution itself. So, the differential equation is transformed into an integral equation. That way, the problem remains unsolved.
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Re: How to solve this second order Diff. Equation.

    Quote Originally Posted by JJacquelin View Post
    I don't agree with the method proposed by FernandoRevilla because F(x) is defined relatively to V(x) which is not known. i.e. the solution is expressed as a function of the solution itself. So, the differential equation is transformed into an integral equation. That way, the problem remains unsolved.
    I didn't propose a solution, only a theoretical and conditional outline.
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  6. #6
    MHF Contributor chisigma's Avatar
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    Re: How to solve this second order Diff. Equation.

    Quote Originally Posted by Arturo_026 View Post
    I took a course on ODE last year, but I do not remember seeing something of this sort:

    (d^2V/dx^2)= N/sqrt(V) Here V is in terms of x, and N is a constant.

    Thank u for any suggestions!
    A second order ODE written in the form...

    y^{''}=f(y) (1)

    ... is solved via the identity...

    y^{''}= \frac{d y^{'}}{dx}= y^{'}\ \frac{d y^{'}}{d y} (2)

    ... so that the (1) becomes...

    y^{'}\ d y^{'}= f(y)\ dy (3)

    ... where the variables are separated, so that we easily obtain...

    y^{'}= \pm \sqrt{2\ \phi(y)+ c_{1}} (4)

    ... where \phi(*) is a primitive of f(*). The result (4) is in most cases 'easy' to obtain. In You want to find y , then from (4) You obtain...

    dx= \frac{d y}{\pm \sqrt{2\ \phi(y) + c_{1}}} \implies x= \int \frac{d y}{\pm \sqrt{2\ \phi(y)+c_{1}}} + c_{2} (5)

    ... even if the integral in (5) in most cases is hard to solve...

    Kind regards

    \chi \sigma
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    Re: How to solve this second order Diff. Equation.

    Quote Originally Posted by FernandoRevilla View Post
    I didn't propose a solution, only a theoretical and conditional outline.
    Sorry for the misunderstanding.
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  8. #8
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    Re: How to solve this second order Diff. Equation.

    Quote Originally Posted by chisigma View Post
    A second order ODE written in the form...

    y^{''}=f(y) (1)

    ... is solved via the identity...

    y^{''}= \frac{d y^{'}}{dx}= y^{'}\ \frac{d y^{'}}{d y} (2)

    ... so that the (1) becomes...

    y^{'}\ d y^{'}= f(y)\ dy (3)

    ... where the variables are separated, so that we easily obtain...

    y^{'}= \pm \sqrt{2\ \phi(y)+ c_{1}} (4)

    ... where \phi(*) is a primitive of f(*). The result (4) is in most cases 'easy' to obtain. In You want to find y , then from (4) You obtain...

    dx= \frac{d y}{\pm \sqrt{2\ \phi(y) + c_{1}}} \implies x= \int \frac{d y}{\pm \sqrt{2\ \phi(y)+c_{1}}} + c_{2} (5)

    ... even if the integral in (5) in most cases is hard to solve...

    Kind regards

    \chi \sigma
    Yes, in most cases (5) is hard to solve. But, in the present case, Phi(y) is on the form sqrt(y). So, the integral is not too hard.
    Best regards,
    JJ.
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  9. #9
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    Re: How to solve this second order Diff. Equation.

    Thanks JJ for your help! I have two quesrions though.
    1st. I know line two of your attachment is true, but how did you go about coming up with it from the equation in line 1?
    2nd. In order to solve an equation of this sort, what level of math maturity is required? I havent taken any upper divission courses in math yet, but I will next semester. Is this something I was supposed to have solved knowing enough about ODE's?
    This equation, by the way, came up in a (unusually challenging (!)) problem from Electrodynamics where the N is made up of a complicated expression involving many constants.
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