# How to solve this second order Diff. Equation.

• Sep 22nd 2011, 08:28 PM
Arturo_026
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!
• Sep 22nd 2011, 09:18 PM
FernandoRevilla
Re: How to solve this second order Diff. Equation.
Being $F(x)$ an anti-derivative of $\frac{N}{\sqrt{V(x)}}$ then, $V=C_1x+\int F(x)\;dx$ .
• Sep 22nd 2011, 11:50 PM
JJacquelin
Re: How to solve this second order Diff. Equation.
Not so simple... (see attachment)
• Sep 22nd 2011, 11:59 PM
JJacquelin
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.
• Sep 23rd 2011, 12:12 AM
FernandoRevilla
Re: How to solve this second order Diff. Equation.
Quote:

Originally Posted by JJacquelin
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.
• Sep 23rd 2011, 12:17 AM
chisigma
Re: How to solve this second order Diff. Equation.
Quote:

Originally Posted by Arturo_026
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

$\chi$ $\sigma$
• Sep 23rd 2011, 12:29 AM
JJacquelin
Re: How to solve this second order Diff. Equation.
Quote:

Originally Posted by FernandoRevilla
I didn't propose a solution, only a theoretical and conditional outline.

Sorry for the misunderstanding.
• Sep 23rd 2011, 01:13 AM
JJacquelin
Re: How to solve this second order Diff. Equation.
Quote:

Originally Posted by chisigma
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.
• Sep 23rd 2011, 03:11 PM
Arturo_026
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.